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Theorem eqfunfv 5962
Description: Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
Assertion
Ref Expression
eqfunfv  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  =  G  <->  ( dom  F  =  dom  G  /\  A. x  e.  dom  F ( F `  x )  =  ( G `  x ) ) ) )
Distinct variable groups:    x, F    x, G

Proof of Theorem eqfunfv
StepHypRef Expression
1 funfn 5599 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 funfn 5599 . 2  |-  ( Fun 
G  <->  G  Fn  dom  G )
3 eqfnfv2 5958 . 2  |-  ( ( F  Fn  dom  F  /\  G  Fn  dom  G )  ->  ( F  =  G  <->  ( dom  F  =  dom  G  /\  A. x  e.  dom  F ( F `  x )  =  ( G `  x ) ) ) )
41, 2, 3syl2anb 477 1  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  =  G  <->  ( dom  F  =  dom  G  /\  A. x  e.  dom  F ( F `  x )  =  ( G `  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   A.wral 2804   dom cdm 4988   Fun wfun 5564    Fn wfn 5565   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by:  fveqressseq  6003  fnprb  6106  symgfixf1  16661  nodenselem5  29685  comptiunov2i  38216
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