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Theorem f1eq2 5373
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 5305 . . 3  |-  ( A  =  B  ->  ( F : A --> C  <->  F : B
--> C ) )
21anbi1d 461 . 2  |-  ( A  =  B  ->  (
( F : A --> C  /\  Fun  `' F
)  <->  ( F : B
--> C  /\  Fun  `' F ) ) )
3 df-f1 5177 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
4 df-f1 5177 . 2  |-  ( F : B -1-1-> C  <->  ( F : B --> C  /\  Fun  `' F ) )
52, 3, 43bitr4g 222 1  |-  ( A  =  B  ->  ( F : A -1-1-> C  <->  F : B -1-1-> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335   `'ccnv 4587   Fun wfun 5166   -->wf 5168   -1-1->wf1 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-fn 5175  df-f 5176  df-f1 5177
This theorem is referenced by:  f1oeq2  5406  f1eq123d  5409  brdomg  6695  ennnfonelemen  12220
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