fresaun - Metamath Proof Explorer

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Theorem fresaun 5603

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 988	. . . 4 $/\$ $/\$
2		inss1 3591	. . . 4
3		fssres 5599	. . . 4 $/\$
4	1, 2, 3	sylancl 662	. . 3 $/\$ $/\$
5		difss 3504	. . . . 5 $\$
6		fssres 5599	. . . . 5 $/\$ $\$ $\$ $\$
7	1, 5, 6	sylancl 662	. . . 4 $/\$ $/\$ $\$ $\$
8		simp2 989	. . . . 5 $/\$ $/\$
9		difss 3504	. . . . 5 $\$
10		fssres 5599	. . . . 5 $/\$ $\$ $\$ $\$
11	8, 9, 10	sylancl 662	. . . 4 $/\$ $/\$ $\$ $\$
12		indifdir 3627	. . . . . 6 $\$ $\$ $\$ $\$ $\$
13		disjdif 3772	. . . . . . 7 $\$
14	13	difeq1i 3491	. . . . . 6 $\$ $\$ $\$ $\$ $\$
15		0dif 3771	. . . . . 6 $\$ $\$
16	12, 14, 15	3eqtri 2467	. . . . 5 $\$ $\$
17	16	a1i 11	. . . 4 $/\$ $/\$ $\$ $\$
18		fun2 5597	. . . 4 $\$ $\$ $/\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
19	7, 11, 17, 18	syl21anc 1217	. . 3 $/\$ $/\$ $\$ $\$ $\$ $\$
20		indi 3617	. . . . 5 $\$ $\$ $\$ $\$
21		inass 3581	. . . . . . 7 $\$ $\$
22		disjdif 3772	. . . . . . . 8 $\$
23	22	ineq2i 3570	. . . . . . 7 $\$
24		in0 3684	. . . . . . 7
25	21, 23, 24	3eqtri 2467	. . . . . 6 $\$
26		incom 3564	. . . . . . . 8
27	26	ineq1i 3569	. . . . . . 7 $\$ $\$
28		inass 3581	. . . . . . . 8 $\$ $\$
29	13	ineq2i 3570	. . . . . . . 8 $\$
30		in0 3684	. . . . . . . 8
31	28, 29, 30	3eqtri 2467	. . . . . . 7 $\$
32	27, 31	eqtri 2463	. . . . . 6 $\$
33	25, 32	uneq12i 3529	. . . . 5 $\$ $\$
34		un0 3683	. . . . 5
35	20, 33, 34	3eqtri 2467	. . . 4 $\$ $\$
36	35	a1i 11	. . 3 $/\$ $/\$ $\$ $\$
37		fun2 5597	. . 3 $/\$ $\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
38	4, 19, 36, 37	syl21anc 1217	. 2 $/\$ $/\$ $\$ $\$ $\$ $\$
39		ffn 5580	. . . . 5
40		ffn 5580	. . . . 5
41		id 22	. . . . 5
42		resasplit 5602	. . . . 5 $/\$ $/\$ $\$ $\$
43	39, 40, 41, 42	syl3an 1260	. . . 4 $/\$ $/\$ $\$ $\$
44	43	feq1d 5567	. . 3 $/\$ $/\$ $\$ $\$
45		un12 3535	. . . . 5 $\$ $\$ $\$ $\$
46	26	uneq1i 3527	. . . . . . 7 $\$ $\$
47		inundif 3778	. . . . . . 7 $\$
48	46, 47	eqtri 2463	. . . . . 6 $\$
49	48	uneq2i 3528	. . . . 5 $\$ $\$ $\$
50		undif1 3775	. . . . 5 $\$
51	45, 49, 50	3eqtri 2467	. . . 4 $\$ $\$
52	51	feq2i 5573	. . 3 $\$ $\$ $\$ $\$ $\$ $\$
53	44, 52	syl6rbbr 264	. 2 $/\$ $/\$ $\$ $\$ $\$ $\$
54	38, 53	mpbid 210	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ w3a 965

wceq 1369 $\$ cdif 3346

cun 3347

cin 3348

wss 3349

c0 3658

cres 4863

wfn 5434

wf 5435

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1591 ax-4 1602 ax-5 1670 ax-6 1708 ax-7 1728 ax-9 1760 ax-10 1775 ax-11 1780 ax-12 1792 ax-13 1943 ax-ext 2423 ax-sep 4434 ax-nul 4442 ax-pr 4552

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1372 df-ex 1587 df-nf 1590 df-sb 1701 df-eu 2257 df-mo 2258 df-clab 2430 df-cleq 2436 df-clel 2439 df-nfc 2577 df-ne 2622 df-ral 2741 df-rex 2742 df-rab 2745 df-v 2995 df-dif 3352 df-un 3354 df-in 3356 df-ss 3363 df-nul 3659 df-if 3813 df-sn 3899 df-pr 3901 df-op 3905 df-br 4314 df-opab 4372 df-id 4657 df-xp 4867 df-rel 4868 df-cnv 4869 df-co 4870 df-dm 4871 df-rn 4872 df-res 4873 df-fun 5441 df-fn 5442 df-f 5443

This theorem is referenced by: cvmliftlem10 27205 elmapresaun 29135

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