MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1eq1 Structured version   Unicode version

Theorem f1eq1 5635
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 5577 . . 3  |-  ( F  =  G  ->  ( F : A --> B  <->  G : A
--> B ) )
2 cnveq 5047 . . . 4  |-  ( F  =  G  ->  `' F  =  `' G
)
32funeqd 5476 . . 3  |-  ( F  =  G  ->  ( Fun  `' F  <->  Fun  `' G ) )
41, 3anbi12d 693 . 2  |-  ( F  =  G  ->  (
( F : A --> B  /\  Fun  `' F
)  <->  ( G : A
--> B  /\  Fun  `' G ) ) )
5 df-f1 5460 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
6 df-f1 5460 . 2  |-  ( G : A -1-1-> B  <->  ( G : A --> B  /\  Fun  `' G ) )
74, 5, 63bitr4g 281 1  |-  ( F  =  G  ->  ( F : A -1-1-> B  <->  G : A -1-1-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653   `'ccnv 4878   Fun wfun 5449   -->wf 5451   -1-1->wf1 5452
This theorem is referenced by:  f1oeq1  5666  f1eq123d  5670  fun11iun  5696  fo00  5712  tposf12  6505  oacomf1olem  6808  f1dom2g  7126  f1domg  7128  dom3d  7150  domtr  7161  domssex2  7268  1sdom  7312  marypha1lem  7439  fseqenlem1  7906  dfac12lem2  8025  dfac12lem3  8026  ackbij2  8124  fin23lem28  8221  fin23lem32  8225  fin23lem34  8227  fin23lem35  8228  fin23lem41  8233  iundom2g  8416  pwfseqlem5  8539  hashf1lem1  11705  hashf1lem2  11706  hashf1  11707  4sqlem11  13324  conjsubgen  15039  sylow1lem2  15234  sylow2blem1  15255  hauspwpwf1  18020  isuslgra  21373  isusgra  21374  usgrares  21390  sizeusglecusg  21496  2trllemE  21554  constr1trl  21589  specval  23402  zrhchr  24361  qqhre  24387  axlowdim  25901  eldioph2lem2  26820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460
  Copyright terms: Public domain W3C validator