Theorem topjoin 36992

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 23144	. . . . . . 7 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ Top)
2	1	ad2antrl 741	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → 𝑘 ∈ Top)
3		toponmax 23157	. . . . . . . . 9 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑘)
4	3	ad2antrl 741	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → 𝑋 ∈ 𝑘)
5	4	snssd 4750	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → {𝑋} ⊆ 𝑘)
6		simprr 785	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)
7		unissb 4904	. . . . . . . 8 ⊢ (∪ 𝑆 ⊆ 𝑘 ↔ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)
8	6, 7	sylibr 237	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → ∪ 𝑆 ⊆ 𝑘)
9	5, 8	unssd 4141	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → ({𝑋} ∪ ∪ 𝑆) ⊆ 𝑘)
10		tgfiss 23222	. . . . . 6 ⊢ ((𝑘 ∈ Top ∧ ({𝑋} ∪ ∪ 𝑆) ⊆ 𝑘) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘)
11	2, 9, 10	syl2anc 596	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘)
12	11	expr 462	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋)) → (∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘))
13	12	ralrimiva 3156	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ 𝑘))
14		ssintrab 4934	. . 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 237	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ⊆ ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})
16		fibas 23208	. . . . . 6 ⊢ (fi‘({𝑋} ∪ ∪ 𝑆)) ∈ TopBases
17		tgtopon 23202	. . . . . 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 4893	. . . . . . . 8 ⊢ ∪ ({𝑋} ∪ ∪ 𝑆) = (∪ {𝑋} ∪ ∪ ∪ 𝑆)
20		unisng 4888	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
21	20	adantr 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ {𝑋} = 𝑋)
22	21	uneq1d 4117	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (∪ {𝑋} ∪ ∪ ∪ 𝑆) = (𝑋 ∪ ∪ ∪ 𝑆))
23	19, 22	eqtr2id 2810	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 ∪ ∪ ∪ 𝑆) = ∪ ({𝑋} ∪ ∪ 𝑆))
24		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ (TopOn‘𝑋))
25		toponuni 23145	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑘)
26		eqimss2 3993	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ∪ 𝑘 → ∪ 𝑘 ⊆ 𝑋)
27	25, 26	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (TopOn‘𝑋) → ∪ 𝑘 ⊆ 𝑋)
28		sspwuni 5064	. . . . . . . . . . . . . 14 ⊢ (𝑘 ⊆ 𝒫 𝑋 ↔ ∪ 𝑘 ⊆ 𝑋)
29	27, 28	sylibr 237	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
30		velpw 4565	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝒫 𝒫 𝑋 ↔ 𝑘 ⊆ 𝒫 𝑋)
31	29, 30	sylibr 237	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ 𝒫 𝒫 𝑋)
32	31	ssriv 3938	. . . . . . . . . . 11 ⊢ (TopOn‘𝑋) ⊆ 𝒫 𝒫 𝑋
33	24, 32	sstrdi 3946	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝒫 𝑋)
34		sspwuni 5064	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝒫 𝒫 𝑋 ↔ ∪ 𝑆 ⊆ 𝒫 𝑋)
35	33, 34	sylib 221	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ 𝑆 ⊆ 𝒫 𝑋)
36		sspwuni 5064	. . . . . . . . 9 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∪ ∪ 𝑆 ⊆ 𝑋)
37	35, 36	sylib 221	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ ∪ 𝑆 ⊆ 𝑋)
38		ssequn2 4138	. . . . . . . 8 ⊢ (∪ ∪ 𝑆 ⊆ 𝑋 ↔ (𝑋 ∪ ∪ ∪ 𝑆) = 𝑋)
39	37, 38	sylib 221	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 ∪ ∪ ∪ 𝑆) = 𝑋)
40		snex 5408	. . . . . . . . 9 ⊢ {𝑋} ∈ V
41		fvex 6895	. . . . . . . . . . . 12 ⊢ (TopOn‘𝑋) ∈ V
42	41	ssex 5289	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (TopOn‘𝑋) → 𝑆 ∈ V)
43	42	adantl 487	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
44	43	uniexd 7748	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ 𝑆 ∈ V)
45		unexg 7749	. . . . . . . . 9 ⊢ (({𝑋} ∈ V ∧ ∪ 𝑆 ∈ V) → ({𝑋} ∪ ∪ 𝑆) ∈ V)
46	40, 44, 45	sylancr 599	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ ∪ 𝑆) ∈ V)
47		fiuni 9402	. . . . . . . 8 ⊢ (({𝑋} ∪ ∪ 𝑆) ∈ V → ∪ ({𝑋} ∪ ∪ 𝑆) = ∪ (fi‘({𝑋} ∪ ∪ 𝑆)))
48	46, 47	syl 18	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ ({𝑋} ∪ ∪ 𝑆) = ∪ (fi‘({𝑋} ∪ ∪ 𝑆)))
49	23, 39, 48	3eqtr3d 2805	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → 𝑋 = ∪ (fi‘({𝑋} ∪ ∪ 𝑆)))
50	49	fveq2d 6886	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (TopOn‘𝑋) = (TopOn‘∪ (fi‘({𝑋} ∪ ∪ 𝑆))))
51	18, 50	eleqtrrid 2869	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ (TopOn‘𝑋))
52		elssuni 4902	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑆 → 𝑗 ⊆ ∪ 𝑆)
53		ssun2 4128	. . . . . . . 8 ⊢ ∪ 𝑆 ⊆ ({𝑋} ∪ ∪ 𝑆)
54	52, 53	sstrdi 3946	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 → 𝑗 ⊆ ({𝑋} ∪ ∪ 𝑆))
55		ssfii 9393	. . . . . . . 8 ⊢ (({𝑋} ∪ ∪ 𝑆) ∈ V → ({𝑋} ∪ ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ ∪ 𝑆)))
56	46, 55	syl 18	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ ∪ 𝑆)))
57	54, 56	sylan9ssr 3948	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗 ∈ 𝑆) → 𝑗 ⊆ (fi‘({𝑋} ∪ ∪ 𝑆)))
58		bastg 23197	. . . . . . 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 3946	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗 ∈ 𝑆) → 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
61	60	ralrimiva 3156	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑗 ∈ 𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
62		sseq2 3960	. . . . . 6 ⊢ (𝑘 = (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) → (𝑗 ⊆ 𝑘 ↔ 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆)))))
63	62	ralbidv 3187	. . . . 5 ⊢ (𝑘 = (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) → (∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘 ↔ ∀𝑗 ∈ 𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆)))))
64	63	elrab 3648	. . . 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 595	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})
66		intss1 4926	. . 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 18	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘} ⊆ (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))))
68	15, 67	eqssd 3951	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) = ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  𝒫 cpw 4560  {csn 4587  ∪ cuni 4870  ∩ cint 4910  ‘cfv 6537  ficfi 9384  topGenctg 17528  Topctop 23124  TopOnctopon 23141  TopBasesctb 23176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7867  df-1o 8459  df-2o 8460  df-en 8957  df-fin 8960  df-fi 9385  df-topgen 17534  df-top 23125  df-topon 23142  df-bases 23177

This theorem is used by: (None)