Theorem topjoin 36348

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 22806	. . . . . . 7 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ Top)
2	1	ad2antrl 728	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → 𝑘 ∈ Top)
3		toponmax 22819	. . . . . . . . 9 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑘)
4	3	ad2antrl 728	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → 𝑋 ∈ 𝑘)
5	4	snssd 4775	. . . . . . 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 4905	. . . . . . . 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 4157	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → ({𝑋} ∪ ∪ 𝑆) ⊆ 𝑘)
10		tgfiss 22884	. . . . . 6 ⊢ ((𝑘 ∈ Top ∧ ({𝑋} ∪ ∪ 𝑆) ⊆ 𝑘) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘)
11	2, 9, 10	syl2anc 584	. . . . 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 3126	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘))
14		ssintrab 4937	. . 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 22870	. . . . . 6 ⊢ (fi‘({𝑋} ∪ ∪ 𝑆)) ∈ TopBases
17		tgtopon 22864	. . . . . 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 4896	. . . . . . . 8 ⊢ ∪ ({𝑋} ∪ ∪ 𝑆) = (∪ {𝑋} ∪ ∪ ∪ 𝑆)
20		unisng 4891	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
21	20	adantr 480	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ {𝑋} = 𝑋)
22	21	uneq1d 4132	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (∪ {𝑋} ∪ ∪ ∪ 𝑆) = (𝑋 ∪ ∪ ∪ 𝑆))
23	19, 22	eqtr2id 2778	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 ∪ ∪ ∪ 𝑆) = ∪ ({𝑋} ∪ ∪ 𝑆))
24		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ (TopOn‘𝑋))
25		toponuni 22807	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑘)
26		eqimss2 4008	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ∪ 𝑘 → ∪ 𝑘 ⊆ 𝑋)
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (TopOn‘𝑋) → ∪ 𝑘 ⊆ 𝑋)
28		sspwuni 5066	. . . . . . . . . . . . . 14 ⊢ (𝑘 ⊆ 𝒫 𝑋 ↔ ∪ 𝑘 ⊆ 𝑋)
29	27, 28	sylibr 234	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
30		velpw 4570	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝒫 𝒫 𝑋 ↔ 𝑘 ⊆ 𝒫 𝑋)
31	29, 30	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ 𝒫 𝒫 𝑋)
32	31	ssriv 3952	. . . . . . . . . . 11 ⊢ (TopOn‘𝑋) ⊆ 𝒫 𝒫 𝑋
33	24, 32	sstrdi 3961	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝒫 𝑋)
34		sspwuni 5066	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝒫 𝒫 𝑋 ↔ ∪ 𝑆 ⊆ 𝒫 𝑋)
35	33, 34	sylib 218	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ 𝑆 ⊆ 𝒫 𝑋)
36		sspwuni 5066	. . . . . . . . 9 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∪ ∪ 𝑆 ⊆ 𝑋)
37	35, 36	sylib 218	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ ∪ 𝑆 ⊆ 𝑋)
38		ssequn2 4154	. . . . . . . 8 ⊢ (∪ ∪ 𝑆 ⊆ 𝑋 ↔ (𝑋 ∪ ∪ ∪ 𝑆) = 𝑋)
39	37, 38	sylib 218	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 ∪ ∪ ∪ 𝑆) = 𝑋)
40		snex 5393	. . . . . . . . 9 ⊢ {𝑋} ∈ V
41		fvex 6873	. . . . . . . . . . . 12 ⊢ (TopOn‘𝑋) ∈ V
42	41	ssex 5278	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (TopOn‘𝑋) → 𝑆 ∈ V)
43	42	adantl 481	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
44	43	uniexd 7720	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ 𝑆 ∈ V)
45		unexg 7721	. . . . . . . . 9 ⊢ (({𝑋} ∈ V ∧ ∪ 𝑆 ∈ V) → ({𝑋} ∪ ∪ 𝑆) ∈ V)
46	40, 44, 45	sylancr 587	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ ∪ 𝑆) ∈ V)
47		fiuni 9385	. . . . . . . 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 2773	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑋 = ∪ (fi‘({𝑋} ∪ ∪ 𝑆)))
50	49	fveq2d 6864	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (TopOn‘𝑋) = (TopOn‘∪ (fi‘({𝑋} ∪ ∪ 𝑆))))
51	18, 50	eleqtrrid 2836	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ (TopOn‘𝑋))
52		elssuni 4903	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑆 → 𝑗 ⊆ ∪ 𝑆)
53		ssun2 4144	. . . . . . . 8 ⊢ ∪ 𝑆 ⊆ ({𝑋} ∪ ∪ 𝑆)
54	52, 53	sstrdi 3961	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 → 𝑗 ⊆ ({𝑋} ∪ ∪ 𝑆))
55		ssfii 9376	. . . . . . . 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 3963	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗 ∈ 𝑆) → 𝑗 ⊆ (fi‘({𝑋} ∪ ∪ 𝑆)))
58		bastg 22859	. . . . . . 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 3961	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗 ∈ 𝑆) → 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
61	60	ralrimiva 3126	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑗 ∈ 𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
62		sseq2 3975	. . . . . 6 ⊢ (𝑘 = (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) → (𝑗 ⊆ 𝑘 ↔ 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆)))))
63	62	ralbidv 3157	. . . . 5 ⊢ (𝑘 = (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) → (∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 ↔ ∀𝑗 ∈ 𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆)))))
64	63	elrab 3661	. . . 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 583	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})
66		intss1 4929	. . 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 3966	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) = ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450   ∪ cun 3914   ⊆ wss 3916  𝒫 cpw 4565  {csn 4591  ∪ cuni 4873  ∩ cint 4912  ‘cfv 6513  ficfi 9367  topGenctg 17406  Topctop 22786  TopOnctopon 22803  TopBasesctb 22838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-om 7845  df-1o 8436  df-2o 8437  df-en 8921  df-fin 8924  df-fi 9368  df-topgen 17412  df-top 22787  df-topon 22804  df-bases 22839

This theorem is referenced by: (None)