### Nuprl Lemma : bij_inv_wf

`∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[bi:Bij(A;B;f)].`
`  (bij_inv(bi) ∈ {g:B ⟶ A| (∀b:B. ((f (g b)) = b ∈ B)) ∧ (∀a:A. ((g (f a)) = a ∈ A))} )`

Proof

Definitions occuring in Statement :  bij_inv: `bij_inv(bi)` biject: `Bij(A;B;f)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` and: `P ∧ Q` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` bij_inv: `bij_inv(bi)` biject: `Bij(A;B;f)` and: `P ∧ Q` pi2: `snd(t)` surject: `Surj(A;B;f)` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` implies: `P `` Q` prop: `ℙ` exists: `∃x:A. B[x]` cand: `A c∧ B` guard: `{T}` inject: `Inj(A;B;f)` pi1: `fst(t)`
Lemmas referenced :  exists_wf equal_wf pi1_wf all_wf biject_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution productElimination thin dependent_set_memberEquality lambdaEquality applyEquality hypothesisEquality extract_by_obid isectElimination cumulativity functionExtensionality hypothesis lambdaFormation dependent_pairEquality equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination independent_pairFormation productEquality axiomEquality isect_memberEquality because_Cache functionEquality universeEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  B].  \mforall{}[bi:Bij(A;B;f)].
(bij\_inv(bi)  \mmember{}  \{g:B  {}\mrightarrow{}  A|  (\mforall{}b:B.  ((f  (g  b))  =  b))  \mwedge{}  (\mforall{}a:A.  ((g  (f  a))  =  a))\}  )

Date html generated: 2017_04_17-AM-07_46_42
Last ObjectModification: 2017_02_27-PM-04_18_04

Theory : list_1

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