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Theorem funeq 5613
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 3562 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 funss 5612 . . 3  |-  ( B 
C_  A  ->  ( Fun  A  ->  Fun  B ) )
31, 2syl 16 . 2  |-  ( A  =  B  ->  ( Fun  A  ->  Fun  B ) )
4 eqimss 3561 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 funss 5612 . . 3  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )
64, 5syl 16 . 2  |-  ( A  =  B  ->  ( Fun  B  ->  Fun  A ) )
73, 6impbid 191 1  |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    C_ wss 3481   Fun wfun 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-in 3488  df-ss 3495  df-br 4454  df-opab 4512  df-rel 5012  df-cnv 5013  df-co 5014  df-fun 5596
This theorem is referenced by:  funeqi  5614  funeqd  5615  fununi  5660  cnvresid  5664  fneq1  5675  nvof1o  6185  funcnvuni  6748  elpmg  7446  fundmeng  7602  isfsupp  7845  dfac9  8528  axdc3lem2  8843  frlmphllem  18680  locfinreflem  27668  orvcval  28221  elfunsg  29493  bnj1379  33369  bnj1385  33371  bnj1497  33596
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