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	|- (A.f e. A (Fun `'f /\ A.g e. A (f C_ g \/ g C_ f)) -> Fun `'U.A)

Distinct variable group:   f,g,A

Proof of Theorem funcnvuni

Dummy variables x y z w v are mutually distinct and distinct from all other variables.

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

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2614  E.wrex 2615   C_ wss 3257  U.cuni 3891  U_ciun 3969  `'ccnv 4771  Fun 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