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Theorem eqfnfv 5265
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 dffn5im 5219 . . 3  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
2 dffn5im 5219 . . 3  |-  ( G  Fn  A  ->  G  =  ( x  e.  A  |->  ( G `  x ) ) )
31, 2eqeqan12d 2055 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( x  e.  A  |->  ( F `  x
) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
4 funfvex 5192 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
54funfni 4999 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
65ralrimiva 2392 . . . 4  |-  ( F  Fn  A  ->  A. x  e.  A  ( F `  x )  e.  _V )
7 mpteqb 5261 . . . 4  |-  ( 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, 7syl 14 . . 3  |-  ( F  Fn  A  ->  (
( x  e.  A  |->  ( F `  x
) )  =  ( x  e.  A  |->  ( G `  x ) )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
98adantr 261 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( x  e.  A  |->  ( F `  x ) )  =  ( x  e.  A  |->  ( G `  x
) )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
103, 9bitrd 177 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    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306   _Vcvv 2557    |-> cmpt 3818    Fn wfn 4897   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by:  eqfnfv2  5266  eqfnfvd  5268  eqfnfv2f  5269  fvreseq  5271  fneqeql  5275  fconst2g  5376  cocan1  5427  cocan2  5428  tfri3  5953  iser0f  9251
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