Theorem topjoin 36600

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 22903	. . . . . . 7 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ Top)
2	1	ad2antrl 734	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → 𝑘 ∈ Top)
3		toponmax 22916	. . . . . . . . 9 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑘)
4	3	ad2antrl 734	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → 𝑋 ∈ 𝑘)
5	4	snssd 4725	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → {𝑋} ⊆ 𝑘)
6		simprr 778	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)
7		unissb 4878	. . . . . . . 8 ⊢ (∪ 𝑆 ⊆ 𝑘 ↔ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)
8	6, 7	sylibr 235	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → ∪ 𝑆 ⊆ 𝑘)
9	5, 8	unssd 4128	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → ({𝑋} ∪ ∪ 𝑆) ⊆ 𝑘)
10		tgfiss 22981	. . . . . 6 ⊢ ((𝑘 ∈ Top ∧ ({𝑋} ∪ ∪ 𝑆) ⊆ 𝑘) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘)
11	2, 9, 10	syl2anc 590	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘)
12	11	expr 457	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋)) → (∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘))
13	12	ralrimiva 3132	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘))
14		ssintrab 4908	. . 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 235	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})
16		fibas 22967	. . . . . 6 ⊢ (fi‘({𝑋} ∪ ∪ 𝑆)) ∈ TopBases
17		tgtopon 22961	. . . . . 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 4868	. . . . . . . 8 ⊢ ∪ ({𝑋} ∪ ∪ 𝑆) = (∪ {𝑋} ∪ ∪ ∪ 𝑆)
20		unisng 4863	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
21	20	adantr 481	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ {𝑋} = 𝑋)
22	21	uneq1d 4104	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (∪ {𝑋} ∪ ∪ ∪ 𝑆) = (𝑋 ∪ ∪ ∪ 𝑆))
23	19, 22	eqtr2id 2788	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 ∪ ∪ ∪ 𝑆) = ∪ ({𝑋} ∪ ∪ 𝑆))
24		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ (TopOn‘𝑋))
25		toponuni 22904	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑘)
26		eqimss2 3981	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ∪ 𝑘 → ∪ 𝑘 ⊆ 𝑋)
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (TopOn‘𝑋) → ∪ 𝑘 ⊆ 𝑋)
28		sspwuni 5036	. . . . . . . . . . . . . 14 ⊢ (𝑘 ⊆ 𝒫 𝑋 ↔ ∪ 𝑘 ⊆ 𝑋)
29	27, 28	sylibr 235	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
30		velpw 4541	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝒫 𝒫 𝑋 ↔ 𝑘 ⊆ 𝒫 𝑋)
31	29, 30	sylibr 235	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ 𝒫 𝒫 𝑋)
32	31	ssriv 3926	. . . . . . . . . . 11 ⊢ (TopOn‘𝑋) ⊆ 𝒫 𝒫 𝑋
33	24, 32	sstrdi 3934	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝒫 𝑋)
34		sspwuni 5036	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝒫 𝒫 𝑋 ↔ ∪ 𝑆 ⊆ 𝒫 𝑋)
35	33, 34	sylib 219	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ 𝑆 ⊆ 𝒫 𝑋)
36		sspwuni 5036	. . . . . . . . 9 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∪ ∪ 𝑆 ⊆ 𝑋)
37	35, 36	sylib 219	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ ∪ 𝑆 ⊆ 𝑋)
38		ssequn2 4125	. . . . . . . 8 ⊢ (∪ ∪ 𝑆 ⊆ 𝑋 ↔ (𝑋 ∪ ∪ ∪ 𝑆) = 𝑋)
39	37, 38	sylib 219	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 ∪ ∪ ∪ 𝑆) = 𝑋)
40		snex 5375	. . . . . . . . 9 ⊢ {𝑋} ∈ V
41		fvex 6847	. . . . . . . . . . . 12 ⊢ (TopOn‘𝑋) ∈ V
42	41	ssex 5256	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (TopOn‘𝑋) → 𝑆 ∈ V)
43	42	adantl 482	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
44	43	uniexd 7692	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ 𝑆 ∈ V)
45		unexg 7693	. . . . . . . . 9 ⊢ (({𝑋} ∈ V ∧ ∪ 𝑆 ∈ V) → ({𝑋} ∪ ∪ 𝑆) ∈ V)
46	40, 44, 45	sylancr 593	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ ∪ 𝑆) ∈ V)
47		fiuni 9338	. . . . . . . 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 2783	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑋 = ∪ (fi‘({𝑋} ∪ ∪ 𝑆)))
50	49	fveq2d 6838	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (TopOn‘𝑋) = (TopOn‘∪ (fi‘({𝑋} ∪ ∪ 𝑆))))
51	18, 50	eleqtrrid 2847	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ (TopOn‘𝑋))
52		elssuni 4876	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑆 → 𝑗 ⊆ ∪ 𝑆)
53		ssun2 4115	. . . . . . . 8 ⊢ ∪ 𝑆 ⊆ ({𝑋} ∪ ∪ 𝑆)
54	52, 53	sstrdi 3934	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 → 𝑗 ⊆ ({𝑋} ∪ ∪ 𝑆))
55		ssfii 9329	. . . . . . . 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 3936	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗 ∈ 𝑆) → 𝑗 ⊆ (fi‘({𝑋} ∪ ∪ 𝑆)))
58		bastg 22956	. . . . . . 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 3934	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗 ∈ 𝑆) → 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
61	60	ralrimiva 3132	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑗 ∈ 𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
62		sseq2 3948	. . . . . 6 ⊢ (𝑘 = (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) → (𝑗 ⊆ 𝑘 ↔ 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆)))))
63	62	ralbidv 3163	. . . . 5 ⊢ (𝑘 = (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) → (∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 ↔ ∀𝑗 ∈ 𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆)))))
64	63	elrab 3636	. . . 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 589	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})
66		intss1 4900	. . 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 3939	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) = ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {crab 3392  Vcvv 3432   ∪ cun 3888   ⊆ wss 3890  𝒫 cpw 4536  {csn 4562  ∪ cuni 4845  ∩ cint 4884  ‘cfv 6492  ficfi 9320  topGenctg 17398  Topctop 22883  TopOnctopon 22900  TopBasesctb 22935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7814  df-1o 8402  df-2o 8403  df-en 8891  df-fin 8894  df-fi 9321  df-topgen 17404  df-top 22884  df-topon 22901  df-bases 22936

This theorem is referenced by: (None)