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Theorem eqfnfv2f 5628
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 5624 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 5624 . 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 2421 . . . . 5  |-  F/_ x
z
42, 3nffv 5534 . . . 4  |-  F/_ x
( F `  z
)
5 eqfnfv2f.2 . . . . 5  |-  F/_ x G
65, 3nffv 5534 . . . 4  |-  F/_ x
( G `  z
)
74, 6nfeq 2428 . . 3  |-  F/ x
( F `  z
)  =  ( G `
 z )
8 nfv 1607 . . 3  |-  F/ z ( F `  x
)  =  ( G `
 x )
9 fveq2 5527 . . . 4  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
10 fveq2 5527 . . . 4  |-  ( z  =  x  ->  ( G `  z )  =  ( G `  x ) )
119, 10eqeq12d 2299 . . 3  |-  ( z  =  x  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  x )  =  ( G `  x ) ) )
127, 8, 11cbvral 2762 . 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 1625   F/_wnfc 2408   A.wral 2545    Fn wfn 5252   ` cfv 5257
This theorem is referenced by:  trinv  25406  ltrinvlem  25417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-fv 5265
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