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Theorem f1eq1 5428
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 5360 . . 3  |-  ( F  =  G  ->  ( F : A --> B  <->  G : A
--> B ) )
2 cnveq 4813 . . . 4  |-  ( F  =  G  ->  `' F  =  `' G
)
32funeqd 5250 . . 3  |-  ( F  =  G  ->  ( Fun  `' F  <->  Fun  `' G ) )
41, 3anbi12d 473 . 2  |-  ( F  =  G  ->  (
( F : A --> B  /\  Fun  `' F
)  <->  ( G : A
--> B  /\  Fun  `' G ) ) )
5 df-f1 5233 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
6 df-f1 5233 . 2  |-  ( G : A -1-1-> B  <->  ( G : A --> B  /\  Fun  `' G ) )
74, 5, 63bitr4g 223 1  |-  ( F  =  G  ->  ( F : A -1-1-> B  <->  G : A -1-1-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363   `'ccnv 4637   Fun wfun 5222   -->wf 5224   -1-1->wf1 5225
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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233
This theorem is referenced by:  f1oeq1  5461  f1eq123d  5465  fun11iun  5494  fo00  5509  tposf12  6284  f1dom2g  6770  f1domg  6772  dom3d  6788  domtr  6799  djudom  7106  difinfsn  7113  djudoml  7232  djudomr  7233  nninfdc  12468  conjsubgen  13172
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