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Theorem f1eq3 5410
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1eq3  |-  ( A  =  B  ->  ( F : C -1-1-> A  <->  F : C -1-1-> B ) )

Proof of Theorem f1eq3
StepHypRef Expression
1 feq3 5342 . . 3  |-  ( A  =  B  ->  ( F : C --> A  <->  F : C
--> B ) )
21anbi1d 465 . 2  |-  ( A  =  B  ->  (
( F : C --> A  /\  Fun  `' F
)  <->  ( F : C
--> B  /\  Fun  `' F ) ) )
3 df-f1 5213 . 2  |-  ( F : C -1-1-> A  <->  ( F : C --> A  /\  Fun  `' F ) )
4 df-f1 5213 . 2  |-  ( F : C -1-1-> B  <->  ( F : C --> B  /\  Fun  `' F ) )
52, 3, 43bitr4g 223 1  |-  ( A  =  B  ->  ( F : C -1-1-> A  <->  F : C -1-1-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   `'ccnv 4619   Fun wfun 5202   -->wf 5204   -1-1->wf1 5205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-in 3133  df-ss 3140  df-f 5212  df-f1 5213
This theorem is referenced by:  f1oeq3  5443  f1eq123d  5445  tposf12  6260  brdomg  6738
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