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Theorem eqfnfv2f 5588
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 5584 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 5584 . 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 2420 . . . . 5  |-  F/_ x
z
42, 3nffv 5493 . . . 4  |-  F/_ x
( F `  z
)
5 eqfnfv2f.2 . . . . 5  |-  F/_ x G
65, 3nffv 5493 . . . 4  |-  F/_ x
( G `  z
)
74, 6nfeq 2427 . . 3  |-  F/ x
( F `  z
)  =  ( G `
 z )
8 nfv 1605 . . 3  |-  F/ z ( F `  x
)  =  ( G `
 x )
9 fveq2 5486 . . . 4  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
10 fveq2 5486 . . . 4  |-  ( z  =  x  ->  ( G `  z )  =  ( G `  x ) )
119, 10eqeq12d 2298 . . 3  |-  ( z  =  x  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  x )  =  ( G `  x ) ) )
127, 8, 11cbvral 2761 . 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 1623   F/_wnfc 2407   A.wral 2544    Fn wfn 5216   ` cfv 5221
This theorem is referenced by:  trinv  24806  ltrinvlem  24817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-fv 5229
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