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Theorem eqfunfv 5627
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 5283 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 funfn 5283 . 2  |-  ( Fun 
G  <->  G  Fn  dom  G )
3 eqfnfv2 5623 . 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 465 1  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  =  G  <->  ( dom  F  =  dom  G  /\  A. x  e.  dom  F ( F `  x )  =  ( G `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   A.wral 2543   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  nodenselem5  24339  repfuntw  25160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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