fresaun - Metamath Proof Explorer

	Metamath Proof Explorer	< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fresaun		Structured version Unicode version

Theorem fresaun 5768

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 1005	. . . 4 $/\$ $/\$
2		inss1 3682	. . . 4
3		fssres 5763	. . . 4 $/\$
4	1, 2, 3	sylancl 666	. . 3 $/\$ $/\$
5		difss 3592	. . . . 5 $\$
6		fssres 5763	. . . . 5 $/\$ $\$ $\$ $\$
7	1, 5, 6	sylancl 666	. . . 4 $/\$ $/\$ $\$ $\$
8		simp2 1006	. . . . 5 $/\$ $/\$
9		difss 3592	. . . . 5 $\$
10		fssres 5763	. . . . 5 $/\$ $\$ $\$ $\$
11	8, 9, 10	sylancl 666	. . . 4 $/\$ $/\$ $\$ $\$
12		indifdir 3729	. . . . . 6 $\$ $\$ $\$ $\$ $\$
13		disjdif 3867	. . . . . . 7 $\$
14	13	difeq1i 3579	. . . . . 6 $\$ $\$ $\$ $\$ $\$
15		0dif 3866	. . . . . 6 $\$ $\$
16	12, 14, 15	3eqtri 2455	. . . . 5 $\$ $\$
17	16	a1i 11	. . . 4 $/\$ $/\$ $\$ $\$
18		fun2 5761	. . . 4 $\$ $\$ $/\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
19	7, 11, 17, 18	syl21anc 1263	. . 3 $/\$ $/\$ $\$ $\$ $\$ $\$
20		indi 3719	. . . . 5 $\$ $\$ $\$ $\$
21		inass 3672	. . . . . . 7 $\$ $\$
22		disjdif 3867	. . . . . . . 8 $\$
23	22	ineq2i 3661	. . . . . . 7 $\$
24		in0 3788	. . . . . . 7
25	21, 23, 24	3eqtri 2455	. . . . . 6 $\$
26		incom 3655	. . . . . . . 8
27	26	ineq1i 3660	. . . . . . 7 $\$ $\$
28		inass 3672	. . . . . . . 8 $\$ $\$
29	13	ineq2i 3661	. . . . . . . 8 $\$
30		in0 3788	. . . . . . . 8
31	28, 29, 30	3eqtri 2455	. . . . . . 7 $\$
32	27, 31	eqtri 2451	. . . . . 6 $\$
33	25, 32	uneq12i 3618	. . . . 5 $\$ $\$
34		un0 3787	. . . . 5
35	20, 33, 34	3eqtri 2455	. . . 4 $\$ $\$
36	35	a1i 11	. . 3 $/\$ $/\$ $\$ $\$
37		fun2 5761	. . 3 $/\$ $\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
38	4, 19, 36, 37	syl21anc 1263	. 2 $/\$ $/\$ $\$ $\$ $\$ $\$
39		ffn 5743	. . . . 5
40		ffn 5743	. . . . 5
41		id 23	. . . . 5
42		resasplit 5767	. . . . 5 $/\$ $/\$ $\$ $\$
43	39, 40, 41, 42	syl3an 1306	. . . 4 $/\$ $/\$ $\$ $\$
44	43	feq1d 5729	. . 3 $/\$ $/\$ $\$ $\$
45		un12 3624	. . . . 5 $\$ $\$ $\$ $\$
46	26	uneq1i 3616	. . . . . . 7 $\$ $\$
47		inundif 3873	. . . . . . 7 $\$
48	46, 47	eqtri 2451	. . . . . 6 $\$
49	48	uneq2i 3617	. . . . 5 $\$ $\$ $\$
50		undif1 3870	. . . . 5 $\$
51	45, 49, 50	3eqtri 2455	. . . 4 $\$ $\$
52	51	feq2i 5736	. . 3 $\$ $\$ $\$ $\$ $\$ $\$
53	44, 52	syl6rbbr 267	. 2 $/\$ $/\$ $\$ $\$ $\$ $\$
54	38, 53	mpbid 213	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ w3a 982

wceq 1437 $\$ cdif 3433

cun 3434

cin 3435

wss 3436

c0 3761

cres 4852

wfn 5593

wf 5594

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1665 ax-4 1678 ax-5 1748 ax-6 1794 ax-7 1839 ax-9 1872 ax-10 1887 ax-11 1892 ax-12 1905 ax-13 2053 ax-ext 2400 ax-sep 4543 ax-nul 4552 ax-pr 4657

This theorem depends on definitions: df-bi 188 df-or 371 df-an 372 df-3an 984 df-tru 1440 df-ex 1660 df-nf 1664 df-sb 1787 df-eu 2269 df-mo 2270 df-clab 2408 df-cleq 2414 df-clel 2417 df-nfc 2572 df-ne 2620 df-ral 2780 df-rex 2781 df-rab 2784 df-v 3083 df-dif 3439 df-un 3441 df-in 3443 df-ss 3450 df-nul 3762 df-if 3910 df-sn 3997 df-pr 3999 df-op 4003 df-br 4421 df-opab 4480 df-id 4765 df-xp 4856 df-rel 4857 df-cnv 4858 df-co 4859 df-dm 4860 df-rn 4861 df-res 4862 df-fun 5600 df-fn 5601 df-f 5602

This theorem is referenced by: cvmliftlem10 30013 elmapresaun 35532

W3C validator