MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  feq23 Unicode version

Theorem feq23 5520
Description: Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
feq23  |-  ( ( A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  F : C --> D ) )

Proof of Theorem feq23
StepHypRef Expression
1 feq2 5518 . 2  |-  ( A  =  C  ->  ( F : A --> B  <->  F : C
--> B ) )
2 feq3 5519 . 2  |-  ( B  =  D  ->  ( F : C --> B  <->  F : C
--> D ) )
31, 2sylan9bb 681 1  |-  ( ( A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  F : C --> D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649   -->wf 5391
This theorem is referenced by:  feq23i  5528  ismgm  21757  ismndo2  21782  rngomndo  21858  seff  27208
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-in 3271  df-ss 3278  df-fn 5398  df-f 5399
  Copyright terms: Public domain W3C validator