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Theorem eqfunfv 5652
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 5276 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 funfn 5276 . 2  |-  ( Fun 
G  <->  G  Fn  dom  G )
3 eqfnfv2 5648 . 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 291 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    /\ wa 104    <-> wb 105    = wceq 1364   A.wral 2472   dom cdm 4655   Fun wfun 5240    Fn wfn 5241   ` cfv 5246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254
This theorem is referenced by:  ennnfonelemex  12561
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