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Theorem eqfnun 26387
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 4949 . . 3  |-  ( F  =  G  ->  ( F  |`  A )  =  ( G  |`  A ) )
2 reseq1 4949 . . 3  |-  ( F  =  G  ->  ( F  |`  B )  =  ( G  |`  B ) )
31, 2jca 518 . 2  |-  ( F  =  G  ->  (
( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) ) )
4 elun 3316 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
5 fveq1 5524 . . . . . . . . 9  |-  ( ( F  |`  A )  =  ( G  |`  A )  ->  (
( F  |`  A ) `
 x )  =  ( ( G  |`  A ) `  x
) )
6 fvres 5542 . . . . . . . . 9  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
75, 6sylan9req 2336 . . . . . . . 8  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  (
( G  |`  A ) `
 x )  =  ( F `  x
) )
8 fvres 5542 . . . . . . . . 9  |-  ( x  e.  A  ->  (
( G  |`  A ) `
 x )  =  ( G `  x
) )
98adantl 452 . . . . . . . 8  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  (
( G  |`  A ) `
 x )  =  ( G `  x
) )
107, 9eqtr3d 2317 . . . . . . 7  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )
1110adantlr 695 . . . . . 6  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  x  e.  A )  ->  ( F `  x
)  =  ( G `
 x ) )
12 fveq1 5524 . . . . . . . . 9  |-  ( ( F  |`  B )  =  ( G  |`  B )  ->  (
( F  |`  B ) `
 x )  =  ( ( G  |`  B ) `  x
) )
13 fvres 5542 . . . . . . . . 9  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
1412, 13sylan9req 2336 . . . . . . . 8  |-  ( ( ( F  |`  B )  =  ( G  |`  B )  /\  x  e.  B )  ->  (
( G  |`  B ) `
 x )  =  ( F `  x
) )
15 fvres 5542 . . . . . . . . 9  |-  ( x  e.  B  ->  (
( G  |`  B ) `
 x )  =  ( G `  x
) )
1615adantl 452 . . . . . . . 8  |-  ( ( ( F  |`  B )  =  ( G  |`  B )  /\  x  e.  B )  ->  (
( G  |`  B ) `
 x )  =  ( G `  x
) )
1714, 16eqtr3d 2317 . . . . . . 7  |-  ( ( ( F  |`  B )  =  ( G  |`  B )  /\  x  e.  B )  ->  ( F `  x )  =  ( G `  x ) )
1817adantll 694 . . . . . 6  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  x  e.  B )  ->  ( F `  x
)  =  ( G `
 x ) )
1911, 18jaodan 760 . . . . 5  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  ( x  e.  A  \/  x  e.  B
) )  ->  ( F `  x )  =  ( G `  x ) )
204, 19sylan2b 461 . . . 4  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  x  e.  ( A  u.  B ) )  -> 
( F `  x
)  =  ( G `
 x ) )
2120ralrimiva 2626 . . 3  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  ->  A. x  e.  ( A  u.  B
) ( F `  x )  =  ( G `  x ) )
22 eqfnfv 5622 . . 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 212 . 2  |-  ( ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  -> 
( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  ->  F  =  G )
)
243, 23impbid2 195 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150    |` cres 4691    Fn wfn 5250   ` cfv 5255
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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