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Theorem fresaunres1 5685
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 3601 . . 3  |-  ( F  u.  G )  =  ( G  u.  F
)
21reseq1i 5207 . 2  |-  ( ( F  u.  G )  |`  A )  =  ( ( G  u.  F
)  |`  A )
3 incom 3644 . . . . . 6  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43reseq2i 5208 . . . . 5  |-  ( F  |`  ( A  i^i  B
) )  =  ( F  |`  ( B  i^i  A ) )
53reseq2i 5208 . . . . 5  |-  ( G  |`  ( A  i^i  B
) )  =  ( G  |`  ( B  i^i  A ) )
64, 5eqeq12i 2471 . . . 4  |-  ( ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) )  <-> 
( F  |`  ( B  i^i  A ) )  =  ( G  |`  ( B  i^i  A ) ) )
7 eqcom 2460 . . . 4  |-  ( ( F  |`  ( B  i^i  A ) )  =  ( G  |`  ( B  i^i  A ) )  <-> 
( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )
86, 7bitri 249 . . 3  |-  ( ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) )  <-> 
( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )
9 fresaunres2 5684 . . . 4  |-  ( ( G : B --> C  /\  F : A --> C  /\  ( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
1093com12 1192 . . 3  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
118, 10syl3an3b 1257 . 2  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
122, 11syl5eq 2504 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    u. cun 3427    i^i cin 3428    |` cres 4943   -->wf 5515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-dm 4951  df-res 4953  df-fun 5521  df-fn 5522  df-f 5523
This theorem is referenced by:  mapunen  7583  hashf1lem1  12319  ptuncnv  19505  resf1o  26174  cvmliftlem10  27320
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