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Theorem eqfnfv 5474
Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
eqfnfv  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
Distinct variable groups:    x, A    x, F    x, G

Proof of Theorem eqfnfv
StepHypRef Expression
1 dffn5 5420 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 dffn5 5420 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
3 eqeq12 2265 . . 3  |-  ( ( F  =  ( x  e.  A  |->  ( F `
 x ) )  /\  G  =  ( x  e.  A  |->  ( G `  x ) ) )  ->  ( F  =  G  <->  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
41, 2, 3syl2anb 467 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( x  e.  A  |->  ( F `  x
) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
5 fvex 5391 . . . 4  |-  ( F `
 x )  e. 
_V
65rgenw 2572 . . 3  |-  A. x  e.  A  ( F `  x )  e.  _V
7 mpteqb 5466 . . 3  |-  ( A. x  e.  A  ( F `  x )  e.  _V  ->  ( (
x  e.  A  |->  ( F `  x ) )  =  ( x  e.  A  |->  ( G `
 x ) )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
86, 7ax-mp 10 . 2  |-  ( ( x  e.  A  |->  ( F `  x ) )  =  ( x  e.  A  |->  ( G `
 x ) )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
94, 8syl6bb 254 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2509   _Vcvv 2727    e. cmpt 3974    Fn wfn 4587   ` cfv 4592
This theorem is referenced by:  eqfnfv2  5475  eqfnfvd  5477  eqfnfv2f  5478  fvreseq  5480  fndmdifeq0  5483  fneqeql  5485  fconst2g  5580  fnsuppres  5584  cocan1  5653  cocan2  5654  weniso  5704  tfr3  6301  ixpfi2  7038  fipreima  7045  fseqenlem1  7535  fpwwe2lem8  8139  ofsubeq0  9623  ser0f  10977  hashgval2  11238  hashf1lem1  11270  efcvgfsum  12241  prmreclem2  12838  1arithlem4  12847  1arith  12848  isgrpinv  14367  dprdf11  15093  psrbagconf1o  15952  pthaus  17164  xkohaus  17179  cnmpt11  17189  cnmpt21  17197  prdsxmetlem  17764  rolle  19169  tdeglem4  19278  resinf1o  19730  dchrelbas2  20308  dchreq  20329  nmlno0lem  21201  phoeqi  21266  occllem  21712  dfiop2  22163  hoeq  22170  ho01i  22238  hoeq1  22240  kbpj  22366  nmlnop0iALT  22405  lnopco0i  22414  nlelchi  22471  rnbra  22517  kbass5  22530  hmopidmchi  22561  hmopidmpji  22562  pjssdif2i  22584  pjinvari  22601  subfacp1lem3  22884  subfacp1lem5  22886  fprb  23297  rdgprc  23319  eqeefv  23705  axlowdimlem14  23757  surjsec2  24286  repfuntw  24326  cocanfo  25540  eqfnun  25553  sdclem2  25618  rrnmet  25719  rrnequiv  25725  fnnfpeq0  25924  pw2f1ocnv  26296  islindf4  26474  caofcan  26706  addrcom  26847  bnj1542  27578  bnj580  27634  ltrnid  29013  ltrneq2  29026  tendoeq1  29642
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608
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