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Theorem f1eq3 5769
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 5706 . . 3  |-  ( A  =  B  ->  ( F : C --> A  <->  F : C
--> B ) )
21anbi1d 704 . 2  |-  ( A  =  B  ->  (
( F : C --> A  /\  Fun  `' F
)  <->  ( F : C
--> B  /\  Fun  `' F ) ) )
3 df-f1 5584 . 2  |-  ( F : C -1-1-> A  <->  ( F : C --> A  /\  Fun  `' F ) )
4 df-f1 5584 . 2  |-  ( F : C -1-1-> B  <->  ( F : C --> B  /\  Fun  `' F ) )
52, 3, 43bitr4g 288 1  |-  ( A  =  B  ->  ( F : C -1-1-> A  <->  F : C -1-1-> B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374   `'ccnv 4991   Fun wfun 5573   -->wf 5575   -1-1->wf1 5576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-in 3476  df-ss 3483  df-f 5583  df-f1 5584
This theorem is referenced by:  f1oeq3  5800  f1eq123d  5802  tposf12  6970  brdomg  7516  pwfseq  9031  f1linds  18620  isuslgra  24006  isusgra  24007  isusgra0  24010  usgraop  24013  cusgraexilem2  24129  diaf1oN  35802
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