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Theorem eqfnfv2f 5478
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 5474 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
StepHypRef Expression
1 eqfnfv 5474 . 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 2385 . . . . 5  |-  F/_ x
z
42, 3nffv 5384 . . . 4  |-  F/_ x
( F `  z
)
5 eqfnfv2f.2 . . . . 5  |-  F/_ x G
65, 3nffv 5384 . . . 4  |-  F/_ x
( G `  z
)
74, 6nfeq 2392 . . 3  |-  F/ x
( F `  z
)  =  ( G `
 z )
8 nfv 1629 . . 3  |-  F/ z ( F `  x
)  =  ( G `
 x )
9 fveq2 5377 . . . 4  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
10 fveq2 5377 . . . 4  |-  ( z  =  x  ->  ( G `  z )  =  ( G `  x ) )
119, 10eqeq12d 2267 . . 3  |-  ( z  =  x  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  x )  =  ( G `  x ) ) )
127, 8, 11cbvral 2705 . 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 6    <-> wb 178    /\ wa 360    = wceq 1619   F/_wnfc 2372   A.wral 2509    Fn wfn 4587   ` cfv 4592
This theorem is referenced by:  trinv  24561  ltrinvlem  24572
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608
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