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Theorem eqfnfv 5786
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 5731 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 dffn5 5731 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
3 eqeq12 2416 . . 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 466 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( x  e.  A  |->  ( F `  x
) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
5 fvex 5701 . . . 4  |-  ( F `
 x )  e. 
_V
65rgenw 2733 . . 3  |-  A. x  e.  A  ( F `  x )  e.  _V
7 mpteqb 5778 . . 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 8 . 2  |-  ( ( x  e.  A  |->  ( F `  x ) )  =  ( x  e.  A  |->  ( G `
 x ) )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
94, 8syl6bb 253 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 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    e. cmpt 4226    Fn wfn 5408   ` cfv 5413
This theorem is referenced by:  eqfnfv2  5787  eqfnfvd  5789  eqfnfv2f  5790  fvreseq  5792  fndmdifeq0  5795  fneqeql  5797  fconst2g  5905  fnsuppres  5911  cocan1  5983  cocan2  5984  weniso  6034  tfr3  6619  ixpfi2  7363  fipreima  7370  fseqenlem1  7861  fpwwe2lem8  8468  ofsubeq0  9953  ser0f  11331  hashgval2  11607  hashf1lem1  11659  efcvgfsum  12643  prmreclem2  13240  1arithlem4  13249  1arith  13250  isgrpinv  14810  dprdf11  15536  psrbagconf1o  16394  pthaus  17623  xkohaus  17638  cnmpt11  17648  cnmpt21  17656  prdsxmetlem  18351  rolle  19827  tdeglem4  19936  resinf1o  20391  dchrelbas2  20974  dchreq  20995  nmlno0lem  22247  phoeqi  22312  occllem  22758  dfiop2  23209  hoeq  23216  ho01i  23284  hoeq1  23286  kbpj  23412  nmlnop0iALT  23451  lnopco0i  23460  nlelchi  23517  rnbra  23563  kbass5  23576  hmopidmchi  23607  hmopidmpji  23608  pjssdif2i  23630  pjinvari  23647  subfacp1lem3  24821  subfacp1lem5  24823  prodf1f  25173  faclimlem1  25310  fprb  25343  rdgprc  25365  eqeefv  25746  axlowdimlem14  25798  cocanfo  26309  eqfnun  26313  sdclem2  26336  rrnmet  26428  rrnequiv  26434  fnnfpeq0  26629  pw2f1ocnv  26998  islindf4  27176  caofcan  27408  addrcom  27547  bnj1542  28934  bnj580  28990  ltrnid  30617  ltrneq2  30630  tendoeq1  31246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421
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