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Theorem fresaunres1 5617
Description: From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.)
Assertion
Ref Expression
fresaunres1  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  (
( F  u.  G
)  |`  A )  =  F )

Proof of Theorem fresaunres1
StepHypRef Expression
1 uncom 3492 . . 3  |-  ( F  u.  G )  =  ( G  u.  F
)
21reseq1i 5143 . 2  |-  ( ( F  u.  G )  |`  A )  =  ( ( G  u.  F
)  |`  A )
3 incom 3534 . . . . . 6  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43reseq2i 5144 . . . . 5  |-  ( F  |`  ( A  i^i  B
) )  =  ( F  |`  ( B  i^i  A ) )
53reseq2i 5144 . . . . 5  |-  ( G  |`  ( A  i^i  B
) )  =  ( G  |`  ( B  i^i  A ) )
64, 5eqeq12i 2450 . . . 4  |-  ( ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) )  <-> 
( F  |`  ( B  i^i  A ) )  =  ( G  |`  ( B  i^i  A ) ) )
7 eqcom 2439 . . . 4  |-  ( ( F  |`  ( B  i^i  A ) )  =  ( G  |`  ( B  i^i  A ) )  <-> 
( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )
86, 7bitri 242 . . 3  |-  ( ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) )  <-> 
( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )
9 fresaunres2 5616 . . . 4  |-  ( ( G : B --> C  /\  F : A --> C  /\  ( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
1093com12 1158 . . 3  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
118, 10syl3an3b 1223 . 2  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
122, 11syl5eq 2481 1  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  (
( F  u.  G
)  |`  A )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    u. cun 3319    i^i cin 3320    |` cres 4881   -->wf 5451
This theorem is referenced by:  mapunen  7277  hashf1lem1  11705  ptuncnv  17840  cvmliftlem10  24982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-dm 4889  df-res 4891  df-fun 5457  df-fn 5458  df-f 5459
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