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Theorem fresaunres1 5749
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 3641 . . 3  |-  ( F  u.  G )  =  ( G  u.  F
)
21reseq1i 5260 . 2  |-  ( ( F  u.  G )  |`  A )  =  ( ( G  u.  F
)  |`  A )
3 incom 3684 . . . . . 6  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43reseq2i 5261 . . . . 5  |-  ( F  |`  ( A  i^i  B
) )  =  ( F  |`  ( B  i^i  A ) )
53reseq2i 5261 . . . . 5  |-  ( G  |`  ( A  i^i  B
) )  =  ( G  |`  ( B  i^i  A ) )
64, 5eqeq12i 2480 . . . 4  |-  ( ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) )  <-> 
( F  |`  ( B  i^i  A ) )  =  ( G  |`  ( B  i^i  A ) ) )
7 eqcom 2469 . . . 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 5748 . . . 4  |-  ( ( G : B --> C  /\  F : A --> C  /\  ( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
1093com12 1195 . . 3  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
118, 10syl3an3b 1261 . 2  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
122, 11syl5eq 2513 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 968    = wceq 1374    u. cun 3467    i^i cin 3468    |` cres 4994   -->wf 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-dm 5002  df-res 5004  df-fun 5581  df-fn 5582  df-f 5583
This theorem is referenced by:  mapunen  7676  hashf1lem1  12457  ptuncnv  20036  resf1o  27075  cvmliftlem10  28229
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