Theorem tgss2 14258

Description: A criterion for determining whether one topology is finer than another, based on a comparison of their bases. Lemma 2.2 of [Munkres] p. 80. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
tgss2	⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐶,𝑦,𝑧   𝑥,𝑉,𝑦

Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem tgss2

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐵 = ∪ 𝐶)
2		uniexg 4471	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
3	2	adantr 276	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐵 ∈ V)
4	1, 3	eqeltrrd 2271	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐶 ∈ V)
5		uniexb 4505	. . . 4 ⊢ (𝐶 ∈ V ↔ ∪ 𝐶 ∈ V)
6	4, 5	sylibr 134	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → 𝐶 ∈ V)
7		tgss3 14257	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
8	6, 7	syldan 282	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
9		eltg2b 14233	. . . . . . 7 ⊢ (𝐶 ∈ V → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
10	6, 9	syl 14	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
11		elunii 3841	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ ∪ 𝐵)
12	11	ancoms 268	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ ∪ 𝐵)
13		biimt 241	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝐵 → (∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
14	12, 13	syl 14	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦) → (∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
15	14	ralbidva 2490	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝑦 ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝑦 (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
16	10, 15	sylan9bb 462	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
17		ralcom3 2662	. . . . 5 ⊢ (∀𝑥 ∈ 𝑦 (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
18	16, 17	bitrdi 196	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
19	18	ralbidva 2490	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (∀𝑦 ∈ 𝐵 𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
20		dfss3 3170	. . 3 ⊢ (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ (topGen‘𝐶))
21		ralcom 2657	. . 3 ⊢ (∀𝑥 ∈ ∪ 𝐵∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
22	19, 20, 21	3bitr4g 223	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
23	8, 22	bitrd 188	1 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  Vcvv 2760   ⊆ wss 3154  ∪ cuni 3836  ‘cfv 5255  topGenctg 12868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-topgen 12874

This theorem is referenced by:  metss  14673