MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1eq3 Unicode version

Theorem f1eq3 5472
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 5414 . . 3  |-  ( A  =  B  ->  ( F : C --> A  <->  F : C
--> B ) )
21anbi1d 685 . 2  |-  ( A  =  B  ->  (
( F : C --> A  /\  Fun  `' F
)  <->  ( F : C
--> B  /\  Fun  `' F ) ) )
3 df-f1 5297 . 2  |-  ( F : C -1-1-> A  <->  ( F : C --> A  /\  Fun  `' F ) )
4 df-f1 5297 . 2  |-  ( F : C -1-1-> B  <->  ( F : C --> B  /\  Fun  `' F ) )
52, 3, 43bitr4g 279 1  |-  ( A  =  B  ->  ( F : C -1-1-> A  <->  F : C -1-1-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633   `'ccnv 4725   Fun wfun 5286   -->wf 5288   -1-1->wf1 5289
This theorem is referenced by:  f1oeq3  5503  f1eq123d  5505  tposf12  6301  brdomg  6915  pwfseq  8331  f1linds  26443  isuslgra  27324  isusgra  27325  isusgra0  27328  diaf1oN  31138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-in 3193  df-ss 3200  df-f 5296  df-f1 5297
  Copyright terms: Public domain W3C validator