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Theorem eqfnfvd 5487
Description: Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
eqfnfvd.1  |-  ( ph  ->  F  Fn  A )
eqfnfvd.2  |-  ( ph  ->  G  Fn  A )
eqfnfvd.3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )
Assertion
Ref Expression
eqfnfvd  |-  ( ph  ->  F  =  G )
Distinct variable groups:    x, A    x, F    x, G    ph, x

Proof of Theorem eqfnfvd
StepHypRef Expression
1 eqfnfvd.3 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )
21ralrimiva 2480 . 2  |-  ( ph  ->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
3 eqfnfvd.1 . . 3  |-  ( ph  ->  F  Fn  A )
4 eqfnfvd.2 . . 3  |-  ( ph  ->  G  Fn  A )
5 eqfnfv 5484 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
63, 4, 5syl2anc 406 . 2  |-  ( ph  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
72, 6mpbird 166 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   A.wral 2391    Fn wfn 5086   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  foeqcnvco  5657  f1eqcocnv  5658  offeq  5961  tfrlem1  6171  frecrdg  6271  updjudhcoinlf  6931  updjudhcoinrg  6932  seq3val  10182  seqvalcd  10183  seq3feq2  10194  seq3feq  10196  seqfeq3  10236  seq3shft  10561  efcvgfsum  11283  upxp  12347  uptx  12349  dvidlemap  12735  dvrecap  12752  peano4nninf  13034  nninfalllemn  13036  nninfsellemeqinf  13046  nninffeq  13050  refeq  13057
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