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Theorem funeq 5620
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 3517 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 funss 5619 . . 3  |-  ( B 
C_  A  ->  ( Fun  A  ->  Fun  B ) )
31, 2syl 17 . 2  |-  ( A  =  B  ->  ( Fun  A  ->  Fun  B ) )
4 eqimss 3516 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 funss 5619 . . 3  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )
64, 5syl 17 . 2  |-  ( A  =  B  ->  ( Fun  B  ->  Fun  A ) )
73, 6impbid 193 1  |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    C_ wss 3436   Fun wfun 5595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-in 3443  df-ss 3450  df-br 4424  df-opab 4483  df-rel 4860  df-cnv 4861  df-co 4862  df-fun 5603
This theorem is referenced by:  funeqi  5621  funeqd  5622  fununi  5667  cnvresid  5671  fneq1  5682  nvof1o  6194  funcnvuni  6760  elpmg  7498  fundmeng  7654  isfsupp  7896  dfac9  8573  axdc3lem2  8888  frlmphllem  19336  locfinreflem  28675  orvcval  29298  bnj1379  29650  bnj1385  29652  bnj1497  29877  elfunsg  30688  funop  38882  funop1  38883  funsndifnop  38885  fundmge2nop  38889  structvtxvallem  38952  usgredgop  39051
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