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Theorem f1eq1 5432
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1eq1  |-  ( F  =  G  ->  ( F : A -1-1-> B  <->  G : A -1-1-> B ) )

Proof of Theorem f1eq1
StepHypRef Expression
1 feq1 5375 . . 3  |-  ( F  =  G  ->  ( F : A --> B  <->  G : A
--> B ) )
2 cnveq 4855 . . . 4  |-  ( F  =  G  ->  `' F  =  `' G
)
32funeqd 5276 . . 3  |-  ( F  =  G  ->  ( Fun  `' F  <->  Fun  `' G ) )
41, 3anbi12d 691 . 2  |-  ( F  =  G  ->  (
( F : A --> B  /\  Fun  `' F
)  <->  ( G : A
--> B  /\  Fun  `' G ) ) )
5 df-f1 5260 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
6 df-f1 5260 . 2  |-  ( G : A -1-1-> B  <->  ( G : A --> B  /\  Fun  `' G ) )
74, 5, 63bitr4g 279 1  |-  ( F  =  G  ->  ( F : A -1-1-> B  <->  G : A -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:  f1oeq1  5463  f1eq123d  5467  fun11iun  5493  fo00  5509  tposf12  6259  oacomf1olem  6562  f1dom2g  6879  f1domg  6881  dom3d  6903  domtr  6914  domssex2  7021  1sdom  7065  marypha1lem  7186  fseqenlem1  7651  dfac12lem2  7770  dfac12lem3  7771  ackbij2  7869  fin23lem28  7966  fin23lem32  7970  fin23lem34  7972  fin23lem35  7973  fin23lem41  7978  iundom2g  8162  pwfseqlem5  8285  hashf1lem1  11393  hashf1lem2  11394  hashf1  11395  4sqlem11  13002  conjsubgen  14715  sylow1lem2  14910  sylow2blem1  14931  hauspwpwf1  17682  specval  22478  axlowdim  24589  eldioph2lem2  26840  isuslgra  28102  isusgra  28103  usgrares  28115
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-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260
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