MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  feq23i Structured version   Unicode version

Theorem feq23i 5662
Description: Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
feq23i.1  |-  A  =  C
feq23i.2  |-  B  =  D
Assertion
Ref Expression
feq23i  |-  ( F : A --> B  <->  F : C
--> D )

Proof of Theorem feq23i
StepHypRef Expression
1 feq23i.1 . 2  |-  A  =  C
2 feq23i.2 . 2  |-  B  =  D
3 feq23 5653 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  F : C --> D ) )
41, 2, 3mp2an 670 1  |-  ( F : A --> B  <->  F : C
--> D )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1403   -->wf 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-in 3418  df-ss 3425  df-fn 5526  df-f 5527
This theorem is referenced by:  ftpg  6015  funcoppc  15378  cnextfval  20744  uhgra0v  24609  wlkntrllem1  24860  ismgmOLD  25617  elghomOLD  25660  mbfmvolf  28595  eulerpartlemt  28697  tendoset  33742  pwssplit4  35361  lincdifsn  38469
  Copyright terms: Public domain W3C validator