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Theorem eqfnun 26423
Description: Two functions on  A  u.  B are equal if and only if they have equal restrictions to both  A and  B. (Contributed by Jeff Madsen, 19-Jun-2011.)
Assertion
Ref Expression
eqfnun  |-  ( ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  -> 
( F  =  G  <-> 
( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) ) ) )

Proof of Theorem eqfnun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 reseq1 5140 . . 3  |-  ( F  =  G  ->  ( F  |`  A )  =  ( G  |`  A ) )
2 reseq1 5140 . . 3  |-  ( F  =  G  ->  ( F  |`  B )  =  ( G  |`  B ) )
31, 2jca 519 . 2  |-  ( F  =  G  ->  (
( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) ) )
4 elun 3488 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
5 fveq1 5727 . . . . . . . . 9  |-  ( ( F  |`  A )  =  ( G  |`  A )  ->  (
( F  |`  A ) `
 x )  =  ( ( G  |`  A ) `  x
) )
6 fvres 5745 . . . . . . . . 9  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
75, 6sylan9req 2489 . . . . . . . 8  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  (
( G  |`  A ) `
 x )  =  ( F `  x
) )
8 fvres 5745 . . . . . . . . 9  |-  ( x  e.  A  ->  (
( G  |`  A ) `
 x )  =  ( G `  x
) )
98adantl 453 . . . . . . . 8  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  (
( G  |`  A ) `
 x )  =  ( G `  x
) )
107, 9eqtr3d 2470 . . . . . . 7  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )
1110adantlr 696 . . . . . 6  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  x  e.  A )  ->  ( F `  x
)  =  ( G `
 x ) )
12 fveq1 5727 . . . . . . . . 9  |-  ( ( F  |`  B )  =  ( G  |`  B )  ->  (
( F  |`  B ) `
 x )  =  ( ( G  |`  B ) `  x
) )
13 fvres 5745 . . . . . . . . 9  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
1412, 13sylan9req 2489 . . . . . . . 8  |-  ( ( ( F  |`  B )  =  ( G  |`  B )  /\  x  e.  B )  ->  (
( G  |`  B ) `
 x )  =  ( F `  x
) )
15 fvres 5745 . . . . . . . . 9  |-  ( x  e.  B  ->  (
( G  |`  B ) `
 x )  =  ( G `  x
) )
1615adantl 453 . . . . . . . 8  |-  ( ( ( F  |`  B )  =  ( G  |`  B )  /\  x  e.  B )  ->  (
( G  |`  B ) `
 x )  =  ( G `  x
) )
1714, 16eqtr3d 2470 . . . . . . 7  |-  ( ( ( F  |`  B )  =  ( G  |`  B )  /\  x  e.  B )  ->  ( F `  x )  =  ( G `  x ) )
1817adantll 695 . . . . . 6  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  x  e.  B )  ->  ( F `  x
)  =  ( G `
 x ) )
1911, 18jaodan 761 . . . . 5  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  ( x  e.  A  \/  x  e.  B
) )  ->  ( F `  x )  =  ( G `  x ) )
204, 19sylan2b 462 . . . 4  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  x  e.  ( A  u.  B ) )  -> 
( F `  x
)  =  ( G `
 x ) )
2120ralrimiva 2789 . . 3  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  ->  A. x  e.  ( A  u.  B
) ( F `  x )  =  ( G `  x ) )
22 eqfnfv 5827 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  -> 
( F  =  G  <->  A. x  e.  ( A  u.  B )
( F `  x
)  =  ( G `
 x ) ) )
2321, 22syl5ibr 213 . 2  |-  ( ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  -> 
( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  ->  F  =  G )
)
243, 23impbid2 196 1  |-  ( ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  -> 
( F  =  G  <-> 
( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    u. cun 3318    |` cres 4880    Fn wfn 5449   ` cfv 5454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462
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