### Nuprl Lemma : fun-equiv-rel

`∀[X,A:Type]. ∀[E:A ⟶ A ⟶ ℙ].  (EquivRel(A;a,b.E[a;b]) `` EquivRel(X ⟶ A;f,g.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])` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` equiv_rel: `EquivRel(T;x,y.E[x; y])` and: `P ∧ Q` fun-equiv: `fun-equiv(X;a,b.E[a; b];f;g)` refl: `Refl(T;x,y.E[x; y])` all: `∀x:A. B[x]` member: `t ∈ T` cand: `A c∧ B` sym: `Sym(T;x,y.E[x; y])` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` so_apply: `x[s]` trans: `Trans(T;x,y.E[x; y])` so_lambda: `λ2x y.t[x; y]` guard: `{T}`
Lemmas referenced :  all_wf equiv_rel_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation independent_pairFormation sqequalRule hypothesisEquality functionEquality cut because_Cache lemma_by_obid sqequalHypSubstitution isectElimination thin lambdaEquality applyEquality hypothesis productElimination dependent_functionElimination independent_functionElimination cumulativity universeEquality

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

Date html generated: 2016_05_14-AM-06_09_09
Last ObjectModification: 2015_12_26-AM-11_48_14

Theory : quot_1

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