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

Step	Hyp	Ref	Expression
1		uncom 3641	. . 3 |- ( F u. G ) = ( G u. F )
2	1	reseq1i 5260	. 2 |- ( ( F u. G ) |` A ) = ( ( G u. F ) |` A )
3		incom 3684	. . . . . 6 |- ( A i^i B ) = ( B i^i A )
4	3	reseq2i 5261	. . . . 5 |- ( F |` ( A i^i B ) ) = ( F |` ( B i^i A ) )
5	3	reseq2i 5261	. . . . 5 |- ( G |` ( A i^i B ) ) = ( G |` ( B i^i A ) )
6	4, 5	eqeq12i 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 ) ) )
8	6, 7	bitri 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 )
10	9	3com12 1195	. . 3 |- ( ( F : A --> C /\ G : B --> C /\ ( G |` ( B i^i A ) ) = ( F |` ( B i^i A ) ) ) -> ( ( G u. F ) |` A ) = F )
11	8, 10	syl3an3b 1261	. 2 |- ( ( F : A --> C /\ G : B --> C /\ ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) -> ( ( G u. F ) |` A ) = F )
12	2, 11	syl5eq 2513	1 |- ( ( F : A --> C /\ G : B --> C /\ ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) -> ( ( F u. G ) |` A ) = F )

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