MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  foeq2 Structured version   Unicode version

Theorem foeq2 5805
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq2  |-  ( A  =  B  ->  ( F : A -onto-> C  <->  F : B -onto-> C ) )

Proof of Theorem foeq2
StepHypRef Expression
1 fneq2 5681 . . 3  |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
21anbi1d 710 . 2  |-  ( A  =  B  ->  (
( F  Fn  A  /\  ran  F  =  C )  <->  ( F  Fn  B  /\  ran  F  =  C ) ) )
3 df-fo 5605 . 2  |-  ( F : A -onto-> C  <->  ( F  Fn  A  /\  ran  F  =  C ) )
4 df-fo 5605 . 2  |-  ( F : B -onto-> C  <->  ( F  Fn  B  /\  ran  F  =  C ) )
52, 3, 43bitr4g 292 1  |-  ( A  =  B  ->  ( F : A -onto-> C  <->  F : B -onto-> C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438   ran crn 4852    Fn wfn 5594   -onto->wfo 5597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-an 373  df-cleq 2415  df-fn 5602  df-fo 5605
This theorem is referenced by:  f1oeq2  5821  foeq123d  5825  tposfo  7006  brwdom  8086  brwdom2  8092  canthwdom  8098  cfslb2n  8700  fodomg  8955  0ramcl  14974  ghmcyg  17523  txcmpb  20651  qtoptopon  20711  opidon2OLD  26044
  Copyright terms: Public domain W3C validator