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

Theorem funeq 5290
Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
funeq  |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )

Proof of Theorem funeq
StepHypRef Expression
1 eqimss2 3244 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 funss 5289 . . 3  |-  ( B 
C_  A  ->  ( Fun  A  ->  Fun  B ) )
31, 2syl 15 . 2  |-  ( A  =  B  ->  ( Fun  A  ->  Fun  B ) )
4 eqimss 3243 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 funss 5289 . . 3  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )
64, 5syl 15 . 2  |-  ( A  =  B  ->  ( Fun  B  ->  Fun  A ) )
73, 6impbid 183 1  |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    C_ wss 3165   Fun wfun 5265
This theorem is referenced by:  funeqi  5291  funeqd  5292  fununi  5332  funcnvuni  5333  cnvresid  5338  fneq1  5349  elpmg  6802  fundmeng  6951  dfac9  7778  axdc3lem2  8093  nvof1o  23052  measbasedom  23547  orvcval  23673  elfunsg  24526  injrec2  25222  cur1val  25301  isalg  25824  algi  25830  tartarmap  25991  pfsubkl  26150  bnj1379  29179  bnj1385  29181  bnj1497  29406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-in 3172  df-ss 3179  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-fun 5273
  Copyright terms: Public domain W3C validator