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

Theorem funeqd 5591
Description: Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
Hypothesis
Ref Expression
funeqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
funeqd  |-  ( ph  ->  ( Fun  A  <->  Fun  B ) )

Proof of Theorem funeqd
StepHypRef Expression
1 funeqd.1 . 2  |-  ( ph  ->  A  =  B )
2 funeq 5589 . 2  |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
31, 2syl 16 1  |-  ( ph  ->  ( Fun  A  <->  Fun  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-in 3468  df-ss 3475  df-br 4440  df-opab 4498  df-rel 4995  df-cnv 4996  df-co 4997  df-fun 5572
This theorem is referenced by:  funopg  5602  funsng  5616  f1eq1  5758  fvn0ssdmfun  5998  funcnvuni  6726  shftfn  12988  isstruct2  14725  strle1  14815  monfval  15220  ismon  15221  monpropd  15225  isepi  15228  isfth  15402  estrres  15607  lubfun  15809  glbfun  15822  acsficl2d  16005  frlmphl  18983  eengbas  24486  ebtwntg  24487  ecgrtg  24488  elntg  24489  istrl  24741  ispth  24772  isspth  24773  0spth  24775  1pthonlem1  24793  constr2spthlem1  24798  2pthlem1  24799  constr2pth  24805  constr3pthlem2  24858  ajfun  25974  fresf1o  27692  padct  27776  esum2dlem  28321  omssubadd  28508  sitgf  28553  fperdvper  31954  dfateq12d  32453  afvres  32496  usgra2pthspth  32723  fdivval  33414
  Copyright terms: Public domain W3C validator