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

Step	Hyp	Ref	Expression
1		uncom 3495	. . 3 |- ( F u. G ) = ( G u. F )
2	1	reseq1i 5101	. 2 |- ( ( F u. G ) |` A ) = ( ( G u. F ) |` A )
3		incom 3538	. . . . . 6 |- ( A i^i B ) = ( B i^i A )
4	3	reseq2i 5102	. . . . 5 |- ( F |` ( A i^i B ) ) = ( F |` ( B i^i A ) )
5	3	reseq2i 5102	. . . . 5 |- ( G |` ( A i^i B ) ) = ( G |` ( B i^i A ) )
6	4, 5	eqeq12i 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 ) ) )
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 5578	. . . 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 1191	. . 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 1256	. 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 2482	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 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