### Nuprl Lemma : quotient_subtype_quotient

`∀[A,B:Type]. ∀[EA:A ⟶ A ⟶ ℙ]. ∀[EB:B ⟶ B ⟶ ℙ].`
`  ((x,y:A//EA[x;y]) ⊆r (x,y:B//EB[x;y])) supposing `
`     ((∀x,y:A.  (EA[x;y] `` EB[x;y])) and `
`     EquivRel(A;x,y.EA[x;y]) and `
`     EquivRel(B;x,y.EB[x;y]) and `
`     (A ⊆r B))`

Proof

Definitions occuring in Statement :  equiv_rel: `EquivRel(T;x,y.E[x; y])` quotient: `x,y:A//B[x; y]` uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` subtype_rel: `A ⊆r B` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` all: `∀x:A. B[x]` guard: `{T}` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  quotient_wf quotient-member-eq equal-wf-base subtype_rel_self all_wf equiv_rel_wf subtype_rel_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaEquality pointwiseFunctionalityForEquality extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule applyEquality independent_isectElimination hypothesis pertypeElimination productElimination dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination productEquality because_Cache instantiate axiomEquality functionEquality isect_memberEquality cumulativity universeEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[EA:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[EB:B  {}\mrightarrow{}  B  {}\mrightarrow{}  \mBbbP{}].
((x,y:A//EA[x;y])  \msubseteq{}r  (x,y:B//EB[x;y]))  supposing
((\mforall{}x,y:A.    (EA[x;y]  {}\mRightarrow{}  EB[x;y]))  and
EquivRel(A;x,y.EA[x;y])  and
EquivRel(B;x,y.EB[x;y])  and
(A  \msubseteq{}r  B))

Date html generated: 2018_05_21-PM-00_04_41
Last ObjectModification: 2018_05_11-AM-10_50_13

Theory : quot_1

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