Theorem fresaunres1 5685

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 3601	. . 3 |- ( F u. G ) = ( G u. F )
2	1	reseq1i 5207	. 2 |- ( ( F u. G ) |` A ) = ( ( G u. F ) |` A )
3		incom 3644	. . . . . 6 |- ( A i^i B ) = ( B i^i A )
4	3	reseq2i 5208	. . . . 5 |- ( F |` ( A i^i B ) ) = ( F |` ( B i^i A ) )
5	3	reseq2i 5208	. . . . 5 |- ( G |` ( A i^i B ) ) = ( G |` ( B i^i A ) )
6	4, 5	eqeq12i 2471	. . . 4 |- ( ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) <-> ( F |` ( B i^i A ) ) = ( G |` ( B i^i A ) ) )
7		eqcom 2460	. . . 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 5684	. . . 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 1192	. . 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 1257	. 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 2504	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 1370   u. cun 3427   i^i cin 3428   |` cres 4943   -->wf 5515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-dm 4951  df-res 4953  df-fun 5521  df-fn 5522  df-f 5523

This theorem is referenced by:  mapunen  7583  hashf1lem1  12319  ptuncnv  19505  resf1o  26174  cvmliftlem10  27320