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Theorem eqfnfv 5638
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 5584 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 dffn5 5584 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
3 eqeq12 2308 . . 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 465 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( x  e.  A  |->  ( F `  x
) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
5 fvex 5555 . . . 4  |-  ( F `
 x )  e. 
_V
65rgenw 2623 . . 3  |-  A. x  e.  A  ( F `  x )  e.  _V
7 mpteqb 5630 . . 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 252 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    e. cmpt 4093    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  eqfnfv2  5639  eqfnfvd  5641  eqfnfv2f  5642  fvreseq  5644  fndmdifeq0  5647  fneqeql  5649  fconst2g  5744  fnsuppres  5748  cocan1  5817  cocan2  5818  weniso  5868  tfr3  6431  ixpfi2  7170  fipreima  7177  fseqenlem1  7667  fpwwe2lem8  8275  ofsubeq0  9759  ser0f  11115  hashgval2  11376  hashf1lem1  11409  efcvgfsum  12383  prmreclem2  12980  1arithlem4  12989  1arith  12990  isgrpinv  14548  dprdf11  15274  psrbagconf1o  16136  pthaus  17348  xkohaus  17363  cnmpt11  17373  cnmpt21  17381  prdsxmetlem  17948  rolle  19353  tdeglem4  19462  resinf1o  19914  dchrelbas2  20492  dchreq  20513  nmlno0lem  21387  phoeqi  21452  occllem  21898  dfiop2  22349  hoeq  22356  ho01i  22424  hoeq1  22426  kbpj  22552  nmlnop0iALT  22591  lnopco0i  22600  nlelchi  22657  rnbra  22703  kbass5  22716  hmopidmchi  22747  hmopidmpji  22748  pjssdif2i  22770  pjinvari  22787  subfacp1lem3  23728  subfacp1lem5  23730  prodf1f  24166  fprb  24200  rdgprc  24222  eqeefv  24603  axlowdimlem14  24655  surjsec2  25223  cocanfo  26477  eqfnun  26490  sdclem2  26555  rrnmet  26656  rrnequiv  26662  fnnfpeq0  26861  pw2f1ocnv  27233  islindf4  27411  caofcan  27643  addrcom  27783  bnj1542  29205  bnj580  29261  ltrnid  30946  ltrneq2  30959  tendoeq1  31575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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