Theorem tgss2 21731

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 488	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐵 = ∪ 𝐶)
2		uniexg 7478	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
3	2	adantr 484	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐵 ∈ V)
4	1, 3	eqeltrrd 2834	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐶 ∈ V)
5		uniexb 7499	. . . 4 ⊢ (𝐶 ∈ V ↔ ∪ 𝐶 ∈ V)
6	4, 5	sylibr 237	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → 𝐶 ∈ V)
7		tgss3 21730	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
8	6, 7	syldan 594	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
9		eltg2b 21703	. . . . . . 7 ⊢ (𝐶 ∈ V → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
10	6, 9	syl 17	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
11		elunii 4798	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ ∪ 𝐵)
12	11	ancoms 462	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ ∪ 𝐵)
13		biimt 364	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝐵 → (∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
14	12, 13	syl 17	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦) → (∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
15	14	ralbidva 3108	. . . . . 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 513	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
17		ralcom3 3266	. . . . 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 290	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
19	18	ralbidva 3108	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (∀𝑦 ∈ 𝐵 𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
20		dfss3 3863	. . 3 ⊢ (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ (topGen‘𝐶))
21		ralcom 3257	. . 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 317	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
23	8, 22	bitrd 282	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   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2113  ∀wral 3053  ∃wrex 3054  Vcvv 3397   ⊆ wss 3841  ∪ cuni 4793  ‘cfv 6333  topGenctg 16807

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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fv 6341  df-topgen 16813

This theorem is referenced by:  metss  23254  relowlssretop  35146  relowlpssretop  35147