Theorem topjoin 36331

Description: Two equivalent formulations of the join of a collection of topologies. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
topjoin	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) = ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})

Distinct variable groups:   𝑗,𝑘,𝑆   𝑗,𝑉,𝑘   𝑗,𝑋,𝑘

Proof of Theorem topjoin

Step	Hyp	Ref	Expression
1		topontop 22940	. . . . . . 7 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ Top)
2	1	ad2antrl 727	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → 𝑘 ∈ Top)
3		toponmax 22953	. . . . . . . . 9 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑘)
4	3	ad2antrl 727	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → 𝑋 ∈ 𝑘)
5	4	snssd 4834	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → {𝑋} ⊆ 𝑘)
6		simprr 772	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)
7		unissb 4963	. . . . . . . 8 ⊢ (∪ 𝑆 ⊆ 𝑘 ↔ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)
8	6, 7	sylibr 234	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → ∪ 𝑆 ⊆ 𝑘)
9	5, 8	unssd 4215	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → ({𝑋} ∪ ∪ 𝑆) ⊆ 𝑘)
10		tgfiss 23019	. . . . . 6 ⊢ ((𝑘 ∈ Top ∧ ({𝑋} ∪ ∪ 𝑆) ⊆ 𝑘) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘)
11	2, 9, 10	syl2anc 583	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘)
12	11	expr 456	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋)) → (∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘))
13	12	ralrimiva 3152	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘))
14		ssintrab 4995	. . 3 ⊢ ((topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘} ↔ ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘))
15	13, 14	sylibr 234	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})
16		fibas 23005	. . . . . 6 ⊢ (fi‘({𝑋} ∪ ∪ 𝑆)) ∈ TopBases
17		tgtopon 22999	. . . . . 6 ⊢ ((fi‘({𝑋} ∪ ∪ 𝑆)) ∈ TopBases → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ (TopOn‘∪ (fi‘({𝑋} ∪ ∪ 𝑆))))
18	16, 17	ax-mp 5	. . . . 5 ⊢ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ (TopOn‘∪ (fi‘({𝑋} ∪ ∪ 𝑆)))
19		uniun 4954	. . . . . . . 8 ⊢ ∪ ({𝑋} ∪ ∪ 𝑆) = (∪ {𝑋} ∪ ∪ ∪ 𝑆)
20		unisng 4949	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
21	20	adantr 480	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ {𝑋} = 𝑋)
22	21	uneq1d 4190	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (∪ {𝑋} ∪ ∪ ∪ 𝑆) = (𝑋 ∪ ∪ ∪ 𝑆))
23	19, 22	eqtr2id 2793	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 ∪ ∪ ∪ 𝑆) = ∪ ({𝑋} ∪ ∪ 𝑆))
24		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ (TopOn‘𝑋))
25		toponuni 22941	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑘)
26		eqimss2 4068	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ∪ 𝑘 → ∪ 𝑘 ⊆ 𝑋)
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (TopOn‘𝑋) → ∪ 𝑘 ⊆ 𝑋)
28		sspwuni 5123	. . . . . . . . . . . . . 14 ⊢ (𝑘 ⊆ 𝒫 𝑋 ↔ ∪ 𝑘 ⊆ 𝑋)
29	27, 28	sylibr 234	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
30		velpw 4627	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝒫 𝒫 𝑋 ↔ 𝑘 ⊆ 𝒫 𝑋)
31	29, 30	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ 𝒫 𝒫 𝑋)
32	31	ssriv 4012	. . . . . . . . . . 11 ⊢ (TopOn‘𝑋) ⊆ 𝒫 𝒫 𝑋
33	24, 32	sstrdi 4021	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝒫 𝑋)
34		sspwuni 5123	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝒫 𝒫 𝑋 ↔ ∪ 𝑆 ⊆ 𝒫 𝑋)
35	33, 34	sylib 218	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ 𝑆 ⊆ 𝒫 𝑋)
36		sspwuni 5123	. . . . . . . . 9 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∪ ∪ 𝑆 ⊆ 𝑋)
37	35, 36	sylib 218	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ ∪ 𝑆 ⊆ 𝑋)
38		ssequn2 4212	. . . . . . . 8 ⊢ (∪ ∪ 𝑆 ⊆ 𝑋 ↔ (𝑋 ∪ ∪ ∪ 𝑆) = 𝑋)
39	37, 38	sylib 218	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 ∪ ∪ ∪ 𝑆) = 𝑋)
40		snex 5451	. . . . . . . . 9 ⊢ {𝑋} ∈ V
41		fvex 6933	. . . . . . . . . . . 12 ⊢ (TopOn‘𝑋) ∈ V
42	41	ssex 5339	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (TopOn‘𝑋) → 𝑆 ∈ V)
43	42	adantl 481	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
44	43	uniexd 7777	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ 𝑆 ∈ V)
45		unexg 7778	. . . . . . . . 9 ⊢ (({𝑋} ∈ V ∧ ∪ 𝑆 ∈ V) → ({𝑋} ∪ ∪ 𝑆) ∈ V)
46	40, 44, 45	sylancr 586	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ ∪ 𝑆) ∈ V)
47		fiuni 9497	. . . . . . . 8 ⊢ (({𝑋} ∪ ∪ 𝑆) ∈ V → ∪ ({𝑋} ∪ ∪ 𝑆) = ∪ (fi‘({𝑋} ∪ ∪ 𝑆)))
48	46, 47	syl 17	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ ({𝑋} ∪ ∪ 𝑆) = ∪ (fi‘({𝑋} ∪ ∪ 𝑆)))
49	23, 39, 48	3eqtr3d 2788	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑋 = ∪ (fi‘({𝑋} ∪ ∪ 𝑆)))
50	49	fveq2d 6924	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (TopOn‘𝑋) = (TopOn‘∪ (fi‘({𝑋} ∪ ∪ 𝑆))))
51	18, 50	eleqtrrid 2851	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ (TopOn‘𝑋))
52		elssuni 4961	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑆 → 𝑗 ⊆ ∪ 𝑆)
53		ssun2 4202	. . . . . . . 8 ⊢ ∪ 𝑆 ⊆ ({𝑋} ∪ ∪ 𝑆)
54	52, 53	sstrdi 4021	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 → 𝑗 ⊆ ({𝑋} ∪ ∪ 𝑆))
55		ssfii 9488	. . . . . . . 8 ⊢ (({𝑋} ∪ ∪ 𝑆) ∈ V → ({𝑋} ∪ ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ ∪ 𝑆)))
56	46, 55	syl 17	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ ∪ 𝑆)))
57	54, 56	sylan9ssr 4023	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗 ∈ 𝑆) → 𝑗 ⊆ (fi‘({𝑋} ∪ ∪ 𝑆)))
58		bastg 22994	. . . . . . 7 ⊢ ((fi‘({𝑋} ∪ ∪ 𝑆)) ∈ TopBases → (fi‘({𝑋} ∪ ∪ 𝑆)) ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
59	16, 58	ax-mp 5	. . . . . 6 ⊢ (fi‘({𝑋} ∪ ∪ 𝑆)) ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆)))
60	57, 59	sstrdi 4021	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗 ∈ 𝑆) → 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
61	60	ralrimiva 3152	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑗 ∈ 𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
62		sseq2 4035	. . . . . 6 ⊢ (𝑘 = (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) → (𝑗 ⊆ 𝑘 ↔ 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆)))))
63	62	ralbidv 3184	. . . . 5 ⊢ (𝑘 = (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) → (∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 ↔ ∀𝑗 ∈ 𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆)))))
64	63	elrab 3708	. . . 4 ⊢ ((topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘} ↔ ((topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆)))))
65	51, 61, 64	sylanbrc 582	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})
66		intss1 4987	. . 3 ⊢ ((topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘} → ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘} ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
67	65, 66	syl 17	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘} ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
68	15, 67	eqssd 4026	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) = ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931  ∩ cint 4970  ‘cfv 6573  ficfi 9479  topGenctg 17497  Topctop 22920  TopOnctopon 22937  TopBasesctb 22973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-2o 8523  df-en 9004  df-fin 9007  df-fi 9480  df-topgen 17503  df-top 22921  df-topon 22938  df-bases 22974

This theorem is referenced by: (None)