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Theorem fresaun 5746

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 997	. . . 4 $/\$ $/\$
2		inss1 3703	. . . 4
3		fssres 5741	. . . 4 $/\$
4	1, 2, 3	sylancl 662	. . 3 $/\$ $/\$
5		difss 3616	. . . . 5 $\$
6		fssres 5741	. . . . 5 $/\$ $\$ $\$ $\$
7	1, 5, 6	sylancl 662	. . . 4 $/\$ $/\$ $\$ $\$
8		simp2 998	. . . . 5 $/\$ $/\$
9		difss 3616	. . . . 5 $\$
10		fssres 5741	. . . . 5 $/\$ $\$ $\$ $\$
11	8, 9, 10	sylancl 662	. . . 4 $/\$ $/\$ $\$ $\$
12		indifdir 3739	. . . . . 6 $\$ $\$ $\$ $\$ $\$
13		disjdif 3886	. . . . . . 7 $\$
14	13	difeq1i 3603	. . . . . 6 $\$ $\$ $\$ $\$ $\$
15		0dif 3885	. . . . . 6 $\$ $\$
16	12, 14, 15	3eqtri 2476	. . . . 5 $\$ $\$
17	16	a1i 11	. . . 4 $/\$ $/\$ $\$ $\$
18		fun2 5739	. . . 4 $\$ $\$ $/\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
19	7, 11, 17, 18	syl21anc 1228	. . 3 $/\$ $/\$ $\$ $\$ $\$ $\$
20		indi 3729	. . . . 5 $\$ $\$ $\$ $\$
21		inass 3693	. . . . . . 7 $\$ $\$
22		disjdif 3886	. . . . . . . 8 $\$
23	22	ineq2i 3682	. . . . . . 7 $\$
24		in0 3797	. . . . . . 7
25	21, 23, 24	3eqtri 2476	. . . . . 6 $\$
26		incom 3676	. . . . . . . 8
27	26	ineq1i 3681	. . . . . . 7 $\$ $\$
28		inass 3693	. . . . . . . 8 $\$ $\$
29	13	ineq2i 3682	. . . . . . . 8 $\$
30		in0 3797	. . . . . . . 8
31	28, 29, 30	3eqtri 2476	. . . . . . 7 $\$
32	27, 31	eqtri 2472	. . . . . 6 $\$
33	25, 32	uneq12i 3641	. . . . 5 $\$ $\$
34		un0 3796	. . . . 5
35	20, 33, 34	3eqtri 2476	. . . 4 $\$ $\$
36	35	a1i 11	. . 3 $/\$ $/\$ $\$ $\$
37		fun2 5739	. . 3 $/\$ $\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
38	4, 19, 36, 37	syl21anc 1228	. 2 $/\$ $/\$ $\$ $\$ $\$ $\$
39		ffn 5721	. . . . 5
40		ffn 5721	. . . . 5
41		id 22	. . . . 5
42		resasplit 5745	. . . . 5 $/\$ $/\$ $\$ $\$
43	39, 40, 41, 42	syl3an 1271	. . . 4 $/\$ $/\$ $\$ $\$
44	43	feq1d 5707	. . 3 $/\$ $/\$ $\$ $\$
45		un12 3647	. . . . 5 $\$ $\$ $\$ $\$
46	26	uneq1i 3639	. . . . . . 7 $\$ $\$
47		inundif 3892	. . . . . . 7 $\$
48	46, 47	eqtri 2472	. . . . . 6 $\$
49	48	uneq2i 3640	. . . . 5 $\$ $\$ $\$
50		undif1 3889	. . . . 5 $\$
51	45, 49, 50	3eqtri 2476	. . . 4 $\$ $\$
52	51	feq2i 5714	. . 3 $\$ $\$ $\$ $\$ $\$ $\$
53	44, 52	syl6rbbr 264	. 2 $/\$ $/\$ $\$ $\$ $\$ $\$
54	38, 53	mpbid 210	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ w3a 974

wceq 1383 $\$ cdif 3458

cun 3459

cin 3460

wss 3461

c0 3770

cres 4991

wfn 5573

wf 5574

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1605 ax-4 1618 ax-5 1691 ax-6 1734 ax-7 1776 ax-9 1808 ax-10 1823 ax-11 1828 ax-12 1840 ax-13 1985 ax-ext 2421 ax-sep 4558 ax-nul 4566 ax-pr 4676

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 976 df-tru 1386 df-ex 1600 df-nf 1604 df-sb 1727 df-eu 2272 df-mo 2273 df-clab 2429 df-cleq 2435 df-clel 2438 df-nfc 2593 df-ne 2640 df-ral 2798 df-rex 2799 df-rab 2802 df-v 3097 df-dif 3464 df-un 3466 df-in 3468 df-ss 3475 df-nul 3771 df-if 3927 df-sn 4015 df-pr 4017 df-op 4021 df-br 4438 df-opab 4496 df-id 4785 df-xp 4995 df-rel 4996 df-cnv 4997 df-co 4998 df-dm 4999 df-rn 5000 df-res 5001 df-fun 5580 df-fn 5581 df-f 5582

This theorem is referenced by: cvmliftlem10 28612 elmapresaun 30679

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