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Theorem funcnvuni 5161

Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5156 for "single-rooted" definition.) (Contributed by set.mm contributors, 11-Aug-2004.)

**Assertion**
Ref	Expression
funcnvuni	$/\$ $\/$

Distinct variable group:

Proof of Theorem funcnvuni Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnveq 4886	. . . . . . . 8
2	1	eqeq2d 2364	. . . . . . 7
3	2	cbvrexv 2836	. . . . . 6
4		cnveq 4886	. . . . . . . . . . 11
5	4	funeqd 5129	. . . . . . . . . 10
6		sseq1 3292	. . . . . . . . . . . 12
7		sseq2 3293	. . . . . . . . . . . 12
8	6, 7	orbi12d 690	. . . . . . . . . . 11 $\/$ $\/$
9	8	ralbidv 2634	. . . . . . . . . 10 $\/$ $\/$
10	5, 9	anbi12d 691	. . . . . . . . 9 $/\$ $\/$ $/\$ $\/$
11	10	rspcv 2951	. . . . . . . 8 $/\$ $\/$ $/\$ $\/$
12		funeq 5127	. . . . . . . . . 10
13	12	biimprcd 216	. . . . . . . . 9
14		sseq2 3293	. . . . . . . . . . . . . . 15
15		sseq1 3292	. . . . . . . . . . . . . . 15
16	14, 15	orbi12d 690	. . . . . . . . . . . . . 14 $\/$ $\/$
17	16	rspcv 2951	. . . . . . . . . . . . 13 $\/$ $\/$
18		cnvss 4885	. . . . . . . . . . . . . . . 16
19		cnvss 4885	. . . . . . . . . . . . . . . 16
20	18, 19	orim12i 502	. . . . . . . . . . . . . . 15 $\/$ $\/$
21		sseq12 3294	. . . . . . . . . . . . . . . . 17 $/\$
22	21	ancoms 439	. . . . . . . . . . . . . . . 16 $/\$
23		sseq12 3294	. . . . . . . . . . . . . . . 16 $/\$
24	22, 23	orbi12d 690	. . . . . . . . . . . . . . 15 $/\$ $\/$ $\/$
25	20, 24	syl5ibrcom 213	. . . . . . . . . . . . . 14 $\/$ $/\$ $\/$
26	25	exp3a 425	. . . . . . . . . . . . 13 $\/$ $\/$
27	17, 26	syl6com 31	. . . . . . . . . . . 12 $\/$ $\/$
28	27	rexlimdv 2737	. . . . . . . . . . 11 $\/$ $\/$
29	28	com23 72	. . . . . . . . . 10 $\/$ $\/$
30	29	alrimdv 1633	. . . . . . . . 9 $\/$ $\/$
31	13, 30	anim12ii 553	. . . . . . . 8 $/\$ $\/$ $/\$ $\/$
32	11, 31	syl6com 31	. . . . . . 7 $/\$ $\/$ $/\$ $\/$
33	32	rexlimdv 2737	. . . . . 6 $/\$ $\/$ $/\$ $\/$
34	3, 33	syl5bi 208	. . . . 5 $/\$ $\/$ $/\$ $\/$
35	34	alrimiv 1631	. . . 4 $/\$ $\/$ $/\$ $\/$
36		df-ral 2619	. . . . 5 $/\$ $\/$ $/\$ $\/$
37		vex 2862	. . . . . . . 8
38		eqeq1 2359	. . . . . . . . 9
39	38	rexbidv 2635	. . . . . . . 8
40	37, 39	elab 2985	. . . . . . 7
41		eqeq1 2359	. . . . . . . . . 10
42	41	rexbidv 2635	. . . . . . . . 9
43	42	ralab 2997	. . . . . . . 8 $\/$ $\/$
44	43	anbi2i 675	. . . . . . 7 $/\$ $\/$ $/\$ $\/$
45	40, 44	imbi12i 316	. . . . . 6 $/\$ $\/$ $/\$ $\/$
46	45	albii 1566	. . . . 5 $/\$ $\/$ $/\$ $\/$
47	36, 46	bitr2i 241	. . . 4 $/\$ $\/$ $/\$ $\/$
48	35, 47	sylib 188	. . 3 $/\$ $\/$ $/\$ $\/$
49		fununi 5160	. . 3 $/\$ $\/$
50	48, 49	syl 15	. 2 $/\$ $\/$
51		cnvuni 4895	. . . 4
52		vex 2862	. . . . . 6
53	52	cnvex 5102	. . . . 5
54	53	dfiun2 4001	. . . 4
55	51, 54	eqtri 2373	. . 3
56	55	funeqi 5128	. 2
57	50, 56	sylibr 203	1 $/\$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 176 $\/$ wo 357 $/\$ wa 358

wal 1540

wceq 1642

wcel 1710

cab 2339

wral 2614

wrex 2615

wss 3257

cuni 3891

ciun 3969

ccnv 4771

wfun 4775

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-iun 3971 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-swap 4724 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-fun 4789

This theorem is referenced by: fun11uni 5162

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