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Theorem fresaunres1 5414
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 3319 . . 3  |-  ( F  u.  G )  =  ( G  u.  F
)
21reseq1i 4951 . 2  |-  ( ( F  u.  G )  |`  A )  =  ( ( G  u.  F
)  |`  A )
3 incom 3361 . . . . . 6  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43reseq2i 4952 . . . . 5  |-  ( F  |`  ( A  i^i  B
) )  =  ( F  |`  ( B  i^i  A ) )
53reseq2i 4952 . . . . 5  |-  ( G  |`  ( A  i^i  B
) )  =  ( G  |`  ( B  i^i  A ) )
64, 5eqeq12i 2296 . . . 4  |-  ( ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) )  <-> 
( F  |`  ( B  i^i  A ) )  =  ( G  |`  ( B  i^i  A ) ) )
7 eqcom 2285 . . . 4  |-  ( ( F  |`  ( B  i^i  A ) )  =  ( G  |`  ( B  i^i  A ) )  <-> 
( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )
86, 7bitri 240 . . 3  |-  ( ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) )  <-> 
( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )
9 fresaunres2 5413 . . . 4  |-  ( ( G : B --> C  /\  F : A --> C  /\  ( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
1093com12 1155 . . 3  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( G  |`  ( B  i^i  A ) )  =  ( F  |`  ( B  i^i  A ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
118, 10syl3an3b 1220 . 2  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  (
( G  u.  F
)  |`  A )  =  F )
122, 11syl5eq 2327 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 934    = wceq 1623    u. cun 3150    i^i cin 3151    |` cres 4691   -->wf 5251
This theorem is referenced by:  mapunen  7030  hashf1lem1  11393  ptuncnv  17498  cvmliftlem10  23825
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  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-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-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-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dm 4699  df-res 4701  df-fun 5257  df-fn 5258  df-f 5259
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