Theorem fresaunres1 5741

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 3587	. . 3 |- ( F u. G ) = ( G u. F )
2	1	reseq1i 5090	. 2 |- ( ( F u. G ) |` A ) = ( ( G u. F ) |` A )
3		incom 3632	. . . . . 6 |- ( A i^i B ) = ( B i^i A )
4	3	reseq2i 5091	. . . . 5 |- ( F |` ( A i^i B ) ) = ( F |` ( B i^i A ) )
5	3	reseq2i 5091	. . . . 5 |- ( G |` ( A i^i B ) ) = ( G |` ( B i^i A ) )
6	4, 5	eqeq12i 2422	. . . 4 |- ( ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) <-> ( F |` ( B i^i A ) ) = ( G |` ( B i^i A ) ) )
7		eqcom 2411	. . . 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 5740	. . . 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 1201	. . 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 1268	. 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 2455	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 974   = wceq 1405   u. cun 3412   i^i cin 3413   |` cres 4825   -->wf 5565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-dm 4833  df-res 4835  df-fun 5571  df-fn 5572  df-f 5573

This theorem is referenced by:  mapunen  7724  hashf1lem1  12553  ptuncnv  20600  resf1o  28000  cvmliftlem10  29591  aacllem  38860