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Theorem eqfnfv2f 5709
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 5705 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 5705 . 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 2494 . . . . 5  |-  F/_ x
z
42, 3nffv 5615 . . . 4  |-  F/_ x
( F `  z
)
5 eqfnfv2f.2 . . . . 5  |-  F/_ x G
65, 3nffv 5615 . . . 4  |-  F/_ x
( G `  z
)
74, 6nfeq 2501 . . 3  |-  F/ x
( F `  z
)  =  ( G `
 z )
8 nfv 1619 . . 3  |-  F/ z ( F `  x
)  =  ( G `
 x )
9 fveq2 5608 . . . 4  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
10 fveq2 5608 . . . 4  |-  ( z  =  x  ->  ( G `  z )  =  ( G `  x ) )
119, 10eqeq12d 2372 . . 3  |-  ( z  =  x  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  x )  =  ( G `  x ) ) )
127, 8, 11cbvral 2836 . 2  |-  ( A. z  e.  A  ( F `  z )  =  ( G `  z )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
131, 12syl6bb 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 1642   F/_wnfc 2481   A.wral 2619    Fn wfn 5332   ` cfv 5337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345
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