Nuprl Lemma : inv_funs-iff

[A,B:Type]. ∀[f:A ⟶ B]. ∀[g:B ⟶ A].  (InvFuns(A;B;f;g) ⇐⇒ (∀a:A. ((g (f a)) a ∈ A)) ∧ (∀b:B. ((f (g b)) b ∈ B)))


Proof




Definitions occuring in Statement :  inv_funs: InvFuns(A;B;f;g) uall: [x:A]. B[x] all: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q apply: a function: x:A ⟶ B[x] universe: Type equal: t ∈ T
Definitions unfolded in proof :  inv_funs: InvFuns(A;B;f;g) tidentity: Id{T} identity: Id uall: [x:A]. B[x] member: t ∈ T iff: ⇐⇒ Q and: P ∧ Q implies:  Q all: x:A. B[x] prop: rev_implies:  Q compose: g squash: T true: True subtype_rel: A ⊆B uimplies: supposing a guard: {T} so_lambda: λ2x.t[x] so_apply: x[s]
Lemmas referenced :  equal-wf-T-base compose_wf equal_wf squash_wf true_wf iff_weakening_equal all_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut independent_pairFormation lambdaFormation sqequalHypSubstitution productElimination thin hypothesis hypothesisEquality productEquality extract_by_obid isectElimination functionEquality cumulativity functionExtensionality applyEquality baseClosed because_Cache lambdaEquality imageElimination equalityTransitivity equalitySymmetry universeEquality dependent_functionElimination natural_numberEquality imageMemberEquality independent_isectElimination independent_functionElimination independent_pairEquality axiomEquality isect_memberEquality hyp_replacement applyLambdaEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  B].  \mforall{}[g:B  {}\mrightarrow{}  A].
    (InvFuns(A;B;f;g)  \mLeftarrow{}{}\mRightarrow{}  (\mforall{}a:A.  ((g  (f  a))  =  a))  \mwedge{}  (\mforall{}b:B.  ((f  (g  b))  =  b)))



Date html generated: 2017_04_14-AM-07_33_16
Last ObjectModification: 2017_02_27-PM-03_07_13

Theory : fun_1


Home Index