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

Theorem f1eq1 5448
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 5391 . . 3  |-  ( F  =  G  ->  ( F : A --> B  <->  G : A
--> B ) )
2 cnveq 4871 . . . 4  |-  ( F  =  G  ->  `' F  =  `' G
)
32funeqd 5292 . . 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 5276 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
6 df-f1 5276 . 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 1632   `'ccnv 4704   Fun wfun 5265   -->wf 5267   -1-1->wf1 5268
This theorem is referenced by:  f1oeq1  5479  f1eq123d  5483  fun11iun  5509  fo00  5525  tposf12  6275  oacomf1olem  6578  f1dom2g  6895  f1domg  6897  dom3d  6919  domtr  6930  domssex2  7037  1sdom  7081  marypha1lem  7202  fseqenlem1  7667  dfac12lem2  7786  dfac12lem3  7787  ackbij2  7885  fin23lem28  7982  fin23lem32  7986  fin23lem34  7988  fin23lem35  7989  fin23lem41  7994  iundom2g  8178  pwfseqlem5  8301  hashf1lem1  11409  hashf1lem2  11410  hashf1  11411  4sqlem11  13018  conjsubgen  14731  sylow1lem2  14926  sylow2blem1  14947  hauspwpwf1  17698  specval  22494  axlowdim  24661  eldioph2lem2  26943  isuslgra  28234  isusgra  28235  usgrares  28249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276
  Copyright terms: Public domain W3C validator