### Nuprl Lemma : strong-continuity2-half-squash-surject-biject

`∀[T,S,U:Type].`
`  ((U ⊆r ℕ)`
`  `` (∃r:ℕ ⟶ U. ∀x:U. ((r x) = x ∈ U))`
`  `` (∃g:ℕ ⟶ T. Surj(ℕ;T;g))`
`  `` (∃h:S ⟶ U. Bij(S;U;h))`
`  `` (∀F:(ℕ ⟶ T) ⟶ S`
`        ⇃(∃M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (S?)`
`           ∀f:ℕ ⟶ T`
`             ((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (S?))) ∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (S?) supposing ↑isl(M n f))))))`

Proof

Definitions occuring in Statement :  biject: `Bij(A;B;f)` surject: `Surj(A;B;f)` quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` true: `True` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` inl: `inl x` union: `left + right` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` cand: `A c∧ B` squash: `↓T` subtype_rel: `A ⊆r B` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` guard: `{T}`
Lemmas referenced :  strong-continuity2_biject_retract-ext strong-continuity2-half-squash nat_wf subtype_rel_self compose_wf exists_wf biject_wf surject_wf all_wf equal_wf subtype_rel_wf strong-continuity2_wf trivial-quotient-true strong-continuity2_functionality_surject int_seg_wf unit_wf2 subtype_rel_dep_function int_seg_subtype_nat false_wf isect_wf assert_wf isl_wf implies-quotient-true2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin cut introduction extract_by_obid isectElimination hypothesis independent_isectElimination because_Cache independent_pairFormation independent_functionElimination sqequalRule imageMemberEquality hypothesisEquality baseClosed dependent_functionElimination lambdaEquality applyEquality functionExtensionality functionEquality cumulativity universeEquality natural_numberEquality setElimination rename unionEquality productEquality inlEquality

Latex:
\mforall{}[T,S,U:Type].
((U  \msubseteq{}r  \mBbbN{})
{}\mRightarrow{}  (\mexists{}r:\mBbbN{}  {}\mrightarrow{}  U.  \mforall{}x:U.  ((r  x)  =  x))
{}\mRightarrow{}  (\mexists{}g:\mBbbN{}  {}\mrightarrow{}  T.  Surj(\mBbbN{};T;g))
{}\mRightarrow{}  (\mexists{}h:S  {}\mrightarrow{}  U.  Bij(S;U;h))
{}\mRightarrow{}  (\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  T)  {}\mrightarrow{}  S
\00D9(\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  T)  {}\mrightarrow{}  (S?)
\mforall{}f:\mBbbN{}  {}\mrightarrow{}  T
((\mexists{}n:\mBbbN{}.  ((M  n  f)  =  (inl  (F  f))))
\mwedge{}  (\mforall{}n:\mBbbN{}.  (M  n  f)  =  (inl  (F  f))  supposing  \muparrow{}isl(M  n  f))))))

Date html generated: 2019_06_20-PM-03_17_37
Last ObjectModification: 2019_06_20-PM-03_13_04

Theory : continuity

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