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Theorem funeq 5475
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 3403 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 funss 5474 . . 3  |-  ( B 
C_  A  ->  ( Fun  A  ->  Fun  B ) )
31, 2syl 16 . 2  |-  ( A  =  B  ->  ( Fun  A  ->  Fun  B ) )
4 eqimss 3402 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 funss 5474 . . 3  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )
64, 5syl 16 . 2  |-  ( A  =  B  ->  ( Fun  B  ->  Fun  A ) )
73, 6impbid 185 1  |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    C_ wss 3322   Fun wfun 5450
This theorem is referenced by:  funeqi  5476  funeqd  5477  fununi  5519  funcnvuni  5520  cnvresid  5525  fneq1  5536  elpmg  7034  fundmeng  7183  dfac9  8018  axdc3lem2  8333  nvof1o  24042  orvcval  24717  elfunsg  25763  bnj1379  29264  bnj1385  29266  bnj1497  29491
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-in 3329  df-ss 3336  df-br 4215  df-opab 4269  df-rel 4887  df-cnv 4888  df-co 4889  df-fun 5458
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