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Theorem funeq 5437
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 3409 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 funss 5436 . . 3  |-  ( B 
C_  A  ->  ( Fun  A  ->  Fun  B ) )
31, 2syl 16 . 2  |-  ( A  =  B  ->  ( Fun  A  ->  Fun  B ) )
4 eqimss 3408 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 funss 5436 . . 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 1369    C_ wss 3328   Fun wfun 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-in 3335  df-ss 3342  df-br 4293  df-opab 4351  df-rel 4847  df-cnv 4848  df-co 4849  df-fun 5420
This theorem is referenced by:  funeqi  5438  funeqd  5439  fununi  5484  cnvresid  5488  fneq1  5499  nvof1o  5987  funcnvuni  6530  elpmg  7228  fundmeng  7384  isfsupp  7624  dfac9  8305  axdc3lem2  8620  frlmphllem  18205  orvcval  26840  elfunsg  27947  bnj1379  31824  bnj1385  31826  bnj1497  32051
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