Theorem funcnvuni 5187

Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5179 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
funcnvuni	⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ◡∪ 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 4708	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ◡𝑥 = ◡𝑣)
2	1	eqeq2d 2149	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (𝑧 = ◡𝑥 ↔ 𝑧 = ◡𝑣))
3	2	cbvrexv 2653	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 ↔ ∃𝑣 ∈ 𝐴 𝑧 = ◡𝑣)
4		cnveq 4708	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑣 → ◡𝑓 = ◡𝑣)
5	4	funeqd 5140	. . . . . . . . . 10 ⊢ (𝑓 = 𝑣 → (Fun ◡𝑓 ↔ Fun ◡𝑣))
6		sseq1 3115	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑣 → (𝑓 ⊆ 𝑔 ↔ 𝑣 ⊆ 𝑔))
7		sseq2 3116	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑣 → (𝑔 ⊆ 𝑓 ↔ 𝑔 ⊆ 𝑣))
8	6, 7	orbi12d 782	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑣 → ((𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓) ↔ (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣)))
9	8	ralbidv 2435	. . . . . . . . . 10 ⊢ (𝑓 = 𝑣 → (∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓) ↔ ∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣)))
10	5, 9	anbi12d 464	. . . . . . . . 9 ⊢ (𝑓 = 𝑣 → ((Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ↔ (Fun ◡𝑣 ∧ ∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣))))
11	10	rspcv 2780	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐴 → (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun ◡𝑣 ∧ ∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣))))
12		funeq 5138	. . . . . . . . . 10 ⊢ (𝑧 = ◡𝑣 → (Fun 𝑧 ↔ Fun ◡𝑣))
13	12	biimprcd 159	. . . . . . . . 9 ⊢ (Fun ◡𝑣 → (𝑧 = ◡𝑣 → Fun 𝑧))
14		sseq2 3116	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑥 → (𝑣 ⊆ 𝑔 ↔ 𝑣 ⊆ 𝑥))
15		sseq1 3115	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑥 → (𝑔 ⊆ 𝑣 ↔ 𝑥 ⊆ 𝑣))
16	14, 15	orbi12d 782	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑥 → ((𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) ↔ (𝑣 ⊆ 𝑥 ∨ 𝑥 ⊆ 𝑣)))
17	16	rspcv 2780	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) → (𝑣 ⊆ 𝑥 ∨ 𝑥 ⊆ 𝑣)))
18		cnvss 4707	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ⊆ 𝑥 → ◡𝑣 ⊆ ◡𝑥)
19		cnvss 4707	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑣 → ◡𝑥 ⊆ ◡𝑣)
20	18, 19	orim12i 748	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ⊆ 𝑥 ∨ 𝑥 ⊆ 𝑣) → (◡𝑣 ⊆ ◡𝑥 ∨ ◡𝑥 ⊆ ◡𝑣))
21		sseq12 3117	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = ◡𝑣 ∧ 𝑤 = ◡𝑥) → (𝑧 ⊆ 𝑤 ↔ ◡𝑣 ⊆ ◡𝑥))
22	21	ancoms 266	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 = ◡𝑥 ∧ 𝑧 = ◡𝑣) → (𝑧 ⊆ 𝑤 ↔ ◡𝑣 ⊆ ◡𝑥))
23		sseq12 3117	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 = ◡𝑥 ∧ 𝑧 = ◡𝑣) → (𝑤 ⊆ 𝑧 ↔ ◡𝑥 ⊆ ◡𝑣))
24	22, 23	orbi12d 782	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = ◡𝑥 ∧ 𝑧 = ◡𝑣) → ((𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧) ↔ (◡𝑣 ⊆ ◡𝑥 ∨ ◡𝑥 ⊆ ◡𝑣)))
25	20, 24	syl5ibrcom 156	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ⊆ 𝑥 ∨ 𝑥 ⊆ 𝑣) → ((𝑤 = ◡𝑥 ∧ 𝑧 = ◡𝑣) → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))
26	25	expd 256	. . . . . . . . . . . . 13 ⊢ ((𝑣 ⊆ 𝑥 ∨ 𝑥 ⊆ 𝑣) → (𝑤 = ◡𝑥 → (𝑧 = ◡𝑣 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))
27	17, 26	syl6com 35	. . . . . . . . . . . 12 ⊢ (∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) → (𝑥 ∈ 𝐴 → (𝑤 = ◡𝑥 → (𝑧 = ◡𝑣 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))))
28	27	rexlimdv 2546	. . . . . . . . . . 11 ⊢ (∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) → (∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 = ◡𝑣 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))
29	28	com23 78	. . . . . . . . . 10 ⊢ (∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) → (𝑧 = ◡𝑣 → (∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))
30	29	alrimdv 1848	. . . . . . . . 9 ⊢ (∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) → (𝑧 = ◡𝑣 → ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))
31	13, 30	anim12ii 340	. . . . . . . 8 ⊢ ((Fun ◡𝑣 ∧ ∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣)) → (𝑧 = ◡𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))))
32	11, 31	syl6com 35	. . . . . . 7 ⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (𝑣 ∈ 𝐴 → (𝑧 = ◡𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))))
33	32	rexlimdv 2546	. . . . . 6 ⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (∃𝑣 ∈ 𝐴 𝑧 = ◡𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))))
34	3, 33	syl5bi 151	. . . . 5 ⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))))
35	34	alrimiv 1846	. . . 4 ⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))))
36		df-ral 2419	. . . . 5 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)) ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} → (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))
37		vex 2684	. . . . . . . 8 ⊢ 𝑧 ∈ V
38		eqeq1 2144	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 = ◡𝑥 ↔ 𝑧 = ◡𝑥))
39	38	rexbidv 2436	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥))
40	37, 39	elab 2823	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} ↔ ∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥)
41		eqeq1 2144	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 = ◡𝑥 ↔ 𝑤 = ◡𝑥))
42	41	rexbidv 2436	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥))
43	42	ralab 2839	. . . . . . . 8 ⊢ (∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧) ↔ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))
44	43	anbi2i 452	. . . . . . 7 ⊢ ((Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)) ↔ (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))
45	40, 44	imbi12i 238	. . . . . 6 ⊢ ((𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} → (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))) ↔ (∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))))
46	45	albii 1446	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} → (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))) ↔ ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))))
47	36, 46	bitr2i 184	. . . 4 ⊢ (∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))) ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))
48	35, 47	sylib 121	. . 3 ⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))
49		fununi 5186	. . 3 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)) → Fun ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥})
50	48, 49	syl 14	. 2 ⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥})
51		cnvuni 4720	. . . 4 ⊢ ◡∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ◡𝑥
52		vex 2684	. . . . . 6 ⊢ 𝑥 ∈ V
53	52	cnvex 5072	. . . . 5 ⊢ ◡𝑥 ∈ V
54	53	dfiun2 3842	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ◡𝑥 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥}
55	51, 54	eqtri 2158	. . 3 ⊢ ◡∪ 𝐴 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥}
56	55	funeqi 5139	. 2 ⊢ (Fun ◡∪ 𝐴 ↔ Fun ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥})
57	50, 56	sylibr 133	1 ⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ◡∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  ∀wal 1329   = wceq 1331   ∈ wcel 1480  {cab 2123  ∀wral 2414  ∃wrex 2415   ⊆ wss 3066  ∪ cuni 3731  ∪ ciun 3808  ^◡ccnv 4533  Fun wfun 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120

This theorem is referenced by:  fun11uni  5188