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Theorem eqfnfv2f 5834
Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5830 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
Hypotheses
Ref Expression
eqfnfv2f.1  |-  F/_ x F
eqfnfv2f.2  |-  F/_ x G
Assertion
Ref Expression
eqfnfv2f  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    F( x)    G( x)

Proof of Theorem eqfnfv2f
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqfnfv 5830 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) )
2 eqfnfv2f.1 . . . . 5  |-  F/_ x F
3 nfcv 2574 . . . . 5  |-  F/_ x
z
42, 3nffv 5738 . . . 4  |-  F/_ x
( F `  z
)
5 eqfnfv2f.2 . . . . 5  |-  F/_ x G
65, 3nffv 5738 . . . 4  |-  F/_ x
( G `  z
)
74, 6nfeq 2581 . . 3  |-  F/ x
( F `  z
)  =  ( G `
 z )
8 nfv 1630 . . 3  |-  F/ z ( F `  x
)  =  ( G `
 x )
9 fveq2 5731 . . . 4  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
10 fveq2 5731 . . . 4  |-  ( z  =  x  ->  ( G `  z )  =  ( G `  x ) )
119, 10eqeq12d 2452 . . 3  |-  ( z  =  x  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  x )  =  ( G `  x ) ) )
127, 8, 11cbvral 2930 . 2  |-  ( A. z  e.  A  ( F `  z )  =  ( G `  z )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
131, 12syl6bb 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 4    <-> wb 178    /\ wa 360    = wceq 1653   F/_wnfc 2561   A.wral 2707    Fn wfn 5452   ` cfv 5457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465
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