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Theorem fresaun 6075

Description: The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)

**Assertion**
Ref	Expression
fresaun	$/\$ $/\$

**Proof of Theorem fresaun**
Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 $/\$ $/\$
2		inss1 3833	. . . 4
3		fssres 6070	. . . 4 $/\$
4	1, 2, 3	sylancl 694	. . 3 $/\$ $/\$
5		difss 3737	. . . . 5 $\$
6		fssres 6070	. . . . 5 $/\$ $\$ $\$ $\$
7	1, 5, 6	sylancl 694	. . . 4 $/\$ $/\$ $\$ $\$
8		simp2 1062	. . . . 5 $/\$ $/\$
9		difss 3737	. . . . 5 $\$
10		fssres 6070	. . . . 5 $/\$ $\$ $\$ $\$
11	8, 9, 10	sylancl 694	. . . 4 $/\$ $/\$ $\$ $\$
12		indifdir 3883	. . . . . 6 $\$ $\$ $\$ $\$ $\$
13		disjdif 4040	. . . . . . 7 $\$
14	13	difeq1i 3724	. . . . . 6 $\$ $\$ $\$ $\$ $\$
15		0dif 3977	. . . . . 6 $\$ $\$
16	12, 14, 15	3eqtri 2648	. . . . 5 $\$ $\$
17	16	a1i 11	. . . 4 $/\$ $/\$ $\$ $\$
18		fun2 6067	. . . 4 $\$ $\$ $/\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
19	7, 11, 17, 18	syl21anc 1325	. . 3 $/\$ $/\$ $\$ $\$ $\$ $\$
20		indi 3873	. . . . 5 $\$ $\$ $\$ $\$
21		inass 3823	. . . . . . 7 $\$ $\$
22		disjdif 4040	. . . . . . . 8 $\$
23	22	ineq2i 3811	. . . . . . 7 $\$
24		in0 3968	. . . . . . 7
25	21, 23, 24	3eqtri 2648	. . . . . 6 $\$
26		incom 3805	. . . . . . . 8
27	26	ineq1i 3810	. . . . . . 7 $\$ $\$
28		inass 3823	. . . . . . . 8 $\$ $\$
29	13	ineq2i 3811	. . . . . . . 8 $\$
30		in0 3968	. . . . . . . 8
31	28, 29, 30	3eqtri 2648	. . . . . . 7 $\$
32	27, 31	eqtri 2644	. . . . . 6 $\$
33	25, 32	uneq12i 3765	. . . . 5 $\$ $\$
34		un0 3967	. . . . 5
35	20, 33, 34	3eqtri 2648	. . . 4 $\$ $\$
36	35	a1i 11	. . 3 $/\$ $/\$ $\$ $\$
37		fun2 6067	. . 3 $/\$ $\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
38	4, 19, 36, 37	syl21anc 1325	. 2 $/\$ $/\$ $\$ $\$ $\$ $\$
39		ffn 6045	. . . . 5
40		ffn 6045	. . . . 5
41		id 22	. . . . 5
42		resasplit 6074	. . . . 5 $/\$ $/\$ $\$ $\$
43	39, 40, 41, 42	syl3an 1368	. . . 4 $/\$ $/\$ $\$ $\$
44	43	feq1d 6030	. . 3 $/\$ $/\$ $\$ $\$
45		un12 3771	. . . . 5 $\$ $\$ $\$ $\$
46	26	uneq1i 3763	. . . . . . 7 $\$ $\$
47		inundif 4046	. . . . . . 7 $\$
48	46, 47	eqtri 2644	. . . . . 6 $\$
49	48	uneq2i 3764	. . . . 5 $\$ $\$ $\$
50		undif1 4043	. . . . 5 $\$
51	45, 49, 50	3eqtri 2648	. . . 4 $\$ $\$
52	51	feq2i 6037	. . 3 $\$ $\$ $\$ $\$ $\$ $\$
53	44, 52	syl6rbbr 279	. 2 $/\$ $/\$ $\$ $\$ $\$ $\$
54	38, 53	mpbid 222	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ w3a 1037

wceq 1483 $\$ cdif 3571

cun 3572

cin 3573

wss 3574

c0 3915

cres 5116

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-eu 2474 df-mo 2475 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-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-fun 5890 df-fn 5891 df-f 5892

This theorem is referenced by: cvmliftlem10 31276 elmapresaun 37334

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