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Theorem fresaun 5756

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 996	. . . 4 $/\$ $/\$
2		inss1 3718	. . . 4
3		fssres 5751	. . . 4 $/\$
4	1, 2, 3	sylancl 662	. . 3 $/\$ $/\$
5		difss 3631	. . . . 5 $\$
6		fssres 5751	. . . . 5 $/\$ $\$ $\$ $\$
7	1, 5, 6	sylancl 662	. . . 4 $/\$ $/\$ $\$ $\$
8		simp2 997	. . . . 5 $/\$ $/\$
9		difss 3631	. . . . 5 $\$
10		fssres 5751	. . . . 5 $/\$ $\$ $\$ $\$
11	8, 9, 10	sylancl 662	. . . 4 $/\$ $/\$ $\$ $\$
12		indifdir 3754	. . . . . 6 $\$ $\$ $\$ $\$ $\$
13		disjdif 3899	. . . . . . 7 $\$
14	13	difeq1i 3618	. . . . . 6 $\$ $\$ $\$ $\$ $\$
15		0dif 3898	. . . . . 6 $\$ $\$
16	12, 14, 15	3eqtri 2500	. . . . 5 $\$ $\$
17	16	a1i 11	. . . 4 $/\$ $/\$ $\$ $\$
18		fun2 5749	. . . 4 $\$ $\$ $/\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
19	7, 11, 17, 18	syl21anc 1227	. . 3 $/\$ $/\$ $\$ $\$ $\$ $\$
20		indi 3744	. . . . 5 $\$ $\$ $\$ $\$
21		inass 3708	. . . . . . 7 $\$ $\$
22		disjdif 3899	. . . . . . . 8 $\$
23	22	ineq2i 3697	. . . . . . 7 $\$
24		in0 3811	. . . . . . 7
25	21, 23, 24	3eqtri 2500	. . . . . 6 $\$
26		incom 3691	. . . . . . . 8
27	26	ineq1i 3696	. . . . . . 7 $\$ $\$
28		inass 3708	. . . . . . . 8 $\$ $\$
29	13	ineq2i 3697	. . . . . . . 8 $\$
30		in0 3811	. . . . . . . 8
31	28, 29, 30	3eqtri 2500	. . . . . . 7 $\$
32	27, 31	eqtri 2496	. . . . . 6 $\$
33	25, 32	uneq12i 3656	. . . . 5 $\$ $\$
34		un0 3810	. . . . 5
35	20, 33, 34	3eqtri 2500	. . . 4 $\$ $\$
36	35	a1i 11	. . 3 $/\$ $/\$ $\$ $\$
37		fun2 5749	. . 3 $/\$ $\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
38	4, 19, 36, 37	syl21anc 1227	. 2 $/\$ $/\$ $\$ $\$ $\$ $\$
39		ffn 5731	. . . . 5
40		ffn 5731	. . . . 5
41		id 22	. . . . 5
42		resasplit 5755	. . . . 5 $/\$ $/\$ $\$ $\$
43	39, 40, 41, 42	syl3an 1270	. . . 4 $/\$ $/\$ $\$ $\$
44	43	feq1d 5717	. . 3 $/\$ $/\$ $\$ $\$
45		un12 3662	. . . . 5 $\$ $\$ $\$ $\$
46	26	uneq1i 3654	. . . . . . 7 $\$ $\$
47		inundif 3905	. . . . . . 7 $\$
48	46, 47	eqtri 2496	. . . . . 6 $\$
49	48	uneq2i 3655	. . . . 5 $\$ $\$ $\$
50		undif1 3902	. . . . 5 $\$
51	45, 49, 50	3eqtri 2500	. . . 4 $\$ $\$
52	51	feq2i 5724	. . 3 $\$ $\$ $\$ $\$ $\$ $\$
53	44, 52	syl6rbbr 264	. 2 $/\$ $/\$ $\$ $\$ $\$ $\$
54	38, 53	mpbid 210	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ w3a 973

wceq 1379 $\$ cdif 3473

cun 3474

cin 3475

wss 3476

c0 3785

cres 5001

wfn 5583

wf 5584

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1601 ax-4 1612 ax-5 1680 ax-6 1719 ax-7 1739 ax-9 1771 ax-10 1786 ax-11 1791 ax-12 1803 ax-13 1968 ax-ext 2445 ax-sep 4568 ax-nul 4576 ax-pr 4686

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1382 df-ex 1597 df-nf 1600 df-sb 1712 df-eu 2279 df-mo 2280 df-clab 2453 df-cleq 2459 df-clel 2462 df-nfc 2617 df-ne 2664 df-ral 2819 df-rex 2820 df-rab 2823 df-v 3115 df-dif 3479 df-un 3481 df-in 3483 df-ss 3490 df-nul 3786 df-if 3940 df-sn 4028 df-pr 4030 df-op 4034 df-br 4448 df-opab 4506 df-id 4795 df-xp 5005 df-rel 5006 df-cnv 5007 df-co 5008 df-dm 5009 df-rn 5010 df-res 5011 df-fun 5590 df-fn 5591 df-f 5592

This theorem is referenced by: cvmliftlem10 28407 elmapresaun 30336

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