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Theorem fresaunres2 6076

Description: From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.)

**Assertion**
Ref	Expression
fresaunres2	$/\$ $/\$

**Proof of Theorem fresaunres2**
Step	Hyp	Ref	Expression
1		ffn 6045	. . . 4
2		ffn 6045	. . . 4
3		id 22	. . . 4
4		resasplit 6074	. . . 4 $/\$ $/\$ $\$ $\$
5	1, 2, 3, 4	syl3an 1368	. . 3 $/\$ $/\$ $\$ $\$
6	5	reseq1d 5395	. 2 $/\$ $/\$ $\$ $\$
7		resundir 5411	. . 3 $\$ $\$ $\$ $\$
8		inss2 3834	. . . . . 6
9		resabs2 5429	. . . . . 6
10	8, 9	ax-mp 5	. . . . 5
11		resundir 5411	. . . . 5 $\$ $\$ $\$ $\$
12	10, 11	uneq12i 3765	. . . 4 $\$ $\$ $\$ $\$
13		dmres 5419	. . . . . . . . 9 $\$ $\$
14		dmres 5419	. . . . . . . . . . 11 $\$ $\$
15	14	ineq2i 3811	. . . . . . . . . 10 $\$ $\$
16		disjdif 4040	. . . . . . . . . . . 12 $\$
17	16	ineq1i 3810	. . . . . . . . . . 11 $\$
18		inass 3823	. . . . . . . . . . 11 $\$ $\$
19		inss1 3833	. . . . . . . . . . . 12
20		0ss 3972	. . . . . . . . . . . 12
21	19, 20	eqssi 3619	. . . . . . . . . . 11
22	17, 18, 21	3eqtr3i 2652	. . . . . . . . . 10 $\$
23	15, 22	eqtri 2644	. . . . . . . . 9 $\$
24	13, 23	eqtri 2644	. . . . . . . 8 $\$
25		relres 5426	. . . . . . . . 9 $\$
26		reldm0 5343	. . . . . . . . 9 $\$ $\$ $\$
27	25, 26	ax-mp 5	. . . . . . . 8 $\$ $\$
28	24, 27	mpbir 221	. . . . . . 7 $\$
29		difss 3737	. . . . . . . 8 $\$
30		resabs2 5429	. . . . . . . 8 $\$ $\$ $\$
31	29, 30	ax-mp 5	. . . . . . 7 $\$ $\$
32	28, 31	uneq12i 3765	. . . . . 6 $\$ $\$ $\$
33	32	uneq2i 3764	. . . . 5 $\$ $\$ $\$
34		simp3 1063	. . . . . . 7 $/\$ $/\$
35	34	uneq1d 3766	. . . . . 6 $/\$ $/\$ $\$ $\$
36		uncom 3757	. . . . . . . . . 10 $\$ $\$
37		un0 3967	. . . . . . . . . 10 $\$ $\$
38	36, 37	eqtri 2644	. . . . . . . . 9 $\$ $\$
39	38	uneq2i 3764	. . . . . . . 8 $\$ $\$
40		resundi 5410	. . . . . . . . 9 $\$ $\$
41		incom 3805	. . . . . . . . . . . . 13
42	41	uneq1i 3763	. . . . . . . . . . . 12 $\$ $\$
43		inundif 4046	. . . . . . . . . . . 12 $\$
44	42, 43	eqtri 2644	. . . . . . . . . . 11 $\$
45	44	reseq2i 5393	. . . . . . . . . 10 $\$
46		fnresdm 6000	. . . . . . . . . . . 12
47	2, 46	syl 17	. . . . . . . . . . 11
48	47	adantl 482	. . . . . . . . . 10 $/\$
49	45, 48	syl5eq 2668	. . . . . . . . 9 $/\$ $\$
50	40, 49	syl5eqr 2670	. . . . . . . 8 $/\$ $\$
51	39, 50	syl5eq 2668	. . . . . . 7 $/\$ $\$
52	51	3adant3 1081	. . . . . 6 $/\$ $/\$ $\$
53	35, 52	eqtrd 2656	. . . . 5 $/\$ $/\$ $\$
54	33, 53	syl5eq 2668	. . . 4 $/\$ $/\$ $\$ $\$
55	12, 54	syl5eq 2668	. . 3 $/\$ $/\$ $\$ $\$
56	7, 55	syl5eq 2668	. 2 $/\$ $/\$ $\$ $\$
57	6, 56	eqtrd 2656	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483 $\$ cdif 3571

cun 3572

cin 3573

wss 3574

c0 3915

cdm 5114

cres 5116

wrel 5119

wfn 5883

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: fresaunres1 6077 mapunen 8129 ptuncnv 21610 cvmliftlem10 31276 elmapresaunres2 37335

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