### Nuprl Lemma : strong-continuity3_functionality_surject

`∀[T,S:Type].`
`  ∀g:T ⟶ S. (Surj(T;S;g) `` (∀F:(ℕ ⟶ S) ⟶ ℕ. (strong-continuity3(T;λf.(F (g o f))) `` strong-continuity3(S;F))))`

Proof

Definitions occuring in Statement :  strong-continuity3: `strong-continuity3(T;F)` surject: `Surj(A;B;f)` compose: `f o g` nat: `ℕ` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  compose: `f o g` rev_implies: `P `` Q` guard: `{T}` true: `True` squash: `↓T` cand: `A c∧ B` not: `¬A` false: `False` less_than': `less_than'(a;b)` le: `A ≤ B` uimplies: `b supposing a` so_apply: `x[s]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` prop: `ℙ` nat: `ℕ` and: `P ∧ Q` iff: `P `⇐⇒` Q` member: `t ∈ T` exists: `∃x:A. B[x]` strong-continuity3: `strong-continuity3(T;F)` implies: `P `` Q` all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]`
Lemmas referenced :  iff_weakening_equal true_wf squash_wf surject_wf strong-continuity3_wf isl_wf assert_wf false_wf int_seg_subtype_nat subtype_rel_dep_function unit_wf2 equal_wf exists_wf all_wf int_seg_wf compose_wf nat_wf surject-inverse
Rules used in proof :  baseClosed imageMemberEquality equalitySymmetry equalityTransitivity imageElimination universeEquality inlEquality independent_pairFormation independent_isectElimination unionEquality productEquality functionEquality because_Cache cumulativity setElimination natural_numberEquality functionExtensionality applyEquality lambdaEquality dependent_pairFormation rename hypothesis independent_functionElimination dependent_functionElimination hypothesisEquality isectElimination extract_by_obid introduction cut sqequalRule thin productElimination sqequalHypSubstitution lambdaFormation isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}[T,S:Type].
\mforall{}g:T  {}\mrightarrow{}  S
(Surj(T;S;g)
{}\mRightarrow{}  (\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  S)  {}\mrightarrow{}  \mBbbN{}.  (strong-continuity3(T;\mlambda{}f.(F  (g  o  f)))  {}\mRightarrow{}  strong-continuity3(S;F))))

Date html generated: 2017_09_29-PM-06_05_10
Last ObjectModification: 2017_09_04-PM-00_14_58

Theory : continuity

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