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Theorem fresaunres1 5579
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 3495 . . 3  |-  ( F  u.  G )  =  ( G  u.  F
)
21reseq1i 5101 . 2  |-  ( ( F  u.  G )  |`  A )  =  ( ( G  u.  F
)  |`  A )
3 incom 3538 . . . . . 6  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43reseq2i 5102 . . . . 5  |-  ( F  |`  ( A  i^i  B
) )  =  ( F  |`  ( B  i^i  A ) )
53reseq2i 5102 . . . . 5  |-  ( G  |`  ( A  i^i  B
) )  =  ( G  |`  ( B  i^i  A ) )
64, 5eqeq12i 2451 . . . 4  |-  ( ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) )  <-> 
( F  |`  ( B  i^i  A ) )  =  ( G  |`  ( B  i^i  A ) ) )
7 eqcom 2440 . . . 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 5578 . . . 4  |-  ( ( G : B --> C  /\  F : A --> C  /\  ( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
1093com12 1191 . . 3  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
118, 10syl3an3b 1256 . 2  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
122, 11syl5eq 2482 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 1369    u. cun 3321    i^i cin 3322    |` cres 4837   -->wf 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-dm 4845  df-res 4847  df-fun 5415  df-fn 5416  df-f 5417
This theorem is referenced by:  mapunen  7472  hashf1lem1  12200  ptuncnv  19355  resf1o  25981  cvmliftlem10  27135
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