Theorem fresaunres1 6077

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 3757	. . 3 |- ( F u. G ) = ( G u. F )
2	1	reseq1i 5392	. 2 |- ( ( F u. G ) |` A ) = ( ( G u. F ) |` A )
3		incom 3805	. . . . . 6 |- ( A i^i B ) = ( B i^i A )
4	3	reseq2i 5393	. . . . 5 |- ( F |` ( A i^i B ) ) = ( F |` ( B i^i A ) )
5	3	reseq2i 5393	. . . . 5 |- ( G |` ( A i^i B ) ) = ( G |` ( B i^i A ) )
6	4, 5	eqeq12i 2636	. . . 4 |- ( ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) <-> ( F |` ( B i^i A ) ) = ( G |` ( B i^i A ) ) )
7		eqcom 2629	. . . 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 264	. . 3 |- ( ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) <-> ( G |` ( B i^i A ) ) = ( F |` ( B i^i A ) ) )
9		fresaunres2 6076	. . . 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 1269	. . 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 1364	. 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 2668	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 1037   = wceq 1483   u. cun 3572   i^i cin 3573   |` cres 5116   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-res 5126  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  mapunen  8129  hashf1lem1  13239  ptuncnv  21610  resf1o  29505  cvmliftlem10  31276  aacllem  42547