Nuprl Lemma : not-quotient-function-subtype

`¬(∀[X,A:Type]. ∀[E:A ⟶ A ⟶ ℙ].`
`    (EquivRel(A;a,b.E[a;b]) `` ((X ⟶ (a,b:A//E[a;b])) ⊆r (f,g:X ⟶ A//fun-equiv(X;a,b.↓E[a;b];f;g)))))`

Proof

Definitions occuring in Statement :  fun-equiv: `fun-equiv(X;a,b.E[a; b];f;g)` equiv_rel: `EquivRel(T;x,y.E[x; y])` quotient: `x,y:A//B[x; y]` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` not: `¬A` squash: `↓T` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  not: `¬A` implies: `P `` Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` uimplies: `b supposing a` subtype_rel: `A ⊆r B` quotient: `x,y:A//B[x; y]` false: `False` and: `P ∧ Q` cand: `A c∧ B` all: `∀x:A. B[x]` true: `True` sq_type: `SQType(T)` guard: `{T}`
Lemmas referenced :  istype-universe equiv_rel_wf subtype_rel_wf quotient_wf fun-equiv_wf base_wf true_wf istype-base equiv_rel_true squash_wf quotient-member-eq subtype_base_sq subtype_rel_self int_subtype_base fun-equiv-rel equiv_rel_squash quotient-squash
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  sqequalRule Error :isectIsType,  cut instantiate introduction extract_by_obid sqequalHypSubstitution isectElimination thin universeEquality hypothesis Error :inhabitedIsType,  hypothesisEquality Error :functionIsType,  Error :universeIsType,  because_Cache Error :lambdaEquality_alt,  applyEquality functionEquality independent_isectElimination independent_functionElimination baseClosed equalityTransitivity equalitySymmetry pertypeElimination promote_hyp productElimination Error :productIsType,  Error :equalityIstype,  sqequalBase dependent_functionElimination natural_numberEquality cumulativity intEquality voidElimination

Latex:
\mneg{}(\mforall{}[X,A:Type].  \mforall{}[E:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].
(EquivRel(A;a,b.E[a;b])
{}\mRightarrow{}  ((X  {}\mrightarrow{}  (a,b:A//E[a;b]))  \msubseteq{}r  (f,g:X  {}\mrightarrow{}  A//fun-equiv(X;a,b.\mdownarrow{}E[a;b];f;g)))))

Date html generated: 2019_06_20-PM-00_32_59
Last ObjectModification: 2018_11_26-AM-00_13_31

Theory : quot_1

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