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Theorem f1eq3 5434
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 5377 . . 3  |-  ( A  =  B  ->  ( F : C --> A  <->  F : C
--> B ) )
21anbi1d 685 . 2  |-  ( A  =  B  ->  (
( F : C --> A  /\  Fun  `' F
)  <->  ( F : C
--> B  /\  Fun  `' F ) ) )
3 df-f1 5260 . 2  |-  ( F : C -1-1-> A  <->  ( F : C --> A  /\  Fun  `' F ) )
4 df-f1 5260 . 2  |-  ( F : C -1-1-> B  <->  ( F : C --> B  /\  Fun  `' F ) )
52, 3, 43bitr4g 279 1  |-  ( A  =  B  ->  ( F : C -1-1-> A  <->  F : C -1-1-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   `'ccnv 4688   Fun wfun 5249   -->wf 5251   -1-1->wf1 5252
This theorem is referenced by:  f1oeq3  5465  f1eq123d  5467  tposf12  6259  brdomg  6872  pwfseq  8286  f1linds  27295  isuslgra  28102  isusgra  28103  isusgra0  28106  diaf1oN  31320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-f 5259  df-f1 5260
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