Theorem fresaunres1 5763

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

Proof of Theorem fresaunres1

Step	Hyp	Ref	Expression
1		uncom 3647	. . 3
2	1	reseq1i 5274	. 2
3		incom 3690	. . . . . 6
4	3	reseq2i 5275	. . . . 5
5	3	reseq2i 5275	. . . . 5
6	4, 5	eqeq12i 2477	. . . 4
7		eqcom 2466	. . . 4
8	6, 7	bitri 249	. . 3
9		fresaunres2 5762	. . . 4
10	9	3com12 1200	. . 3
11	8, 10	syl3an3b 1266	. 2
12	2, 11	syl5eq 2510	1

Syntax hints:  ->wi 4  /\w3a 973  =wceq 1395  u.cun 3473  i^icin 3474  |`cres 5006  -->wf 5589

This theorem is referenced by:  mapunen  7706  hashf1lem1  12504  ptuncnv  20308  resf1o  27553  cvmliftlem10  28739  aacllem  33216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-dm 5014  df-res 5016  df-fun 5595  df-fn 5596  df-f 5597