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Theorem f1eq2 5212
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1eq2  |-  ( A  =  B  ->  ( F : A -1-1-> C  <->  F : B -1-1-> C ) )

Proof of Theorem f1eq2
StepHypRef Expression
1 feq2 5146 . . 3  |-  ( A  =  B  ->  ( F : A --> C  <->  F : B
--> C ) )
21anbi1d 453 . 2  |-  ( A  =  B  ->  (
( F : A --> C  /\  Fun  `' F
)  <->  ( F : B
--> C  /\  Fun  `' F ) ) )
3 df-f1 5020 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
4 df-f1 5020 . 2  |-  ( F : B -1-1-> C  <->  ( F : B --> C  /\  Fun  `' F ) )
52, 3, 43bitr4g 221 1  |-  ( A  =  B  ->  ( F : A -1-1-> C  <->  F : B -1-1-> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   `'ccnv 4437   Fun wfun 5009   -->wf 5011   -1-1->wf1 5012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-fn 5018  df-f 5019  df-f1 5020
This theorem is referenced by:  f1oeq2  5245  f1eq123d  5248  brdomg  6465
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