Theorem tgss2 23144

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 489	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐵 = ∪ 𝐶)
2		uniexg 7738	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
3	2	adantr 485	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐵 ∈ V)
4	1, 3	eqeltrrd 2864	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐶 ∈ V)
5		uniexb 7759	. . . 4 ⊢ (𝐶 ∈ V ↔ ∪ 𝐶 ∈ V)
6	4, 5	sylibr 237	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → 𝐶 ∈ V)
7		tgss3 23143	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
8	6, 7	syldan 602	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
9		eltg2b 23116	. . . . . . 7 ⊢ (𝐶 ∈ V → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
10	6, 9	syl 18	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
11		elunii 4877	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ ∪ 𝐵)
12	11	ancoms 463	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ ∪ 𝐵)
13		biimt 363	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝐵 → (∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
14	12, 13	syl 18	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦) → (∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
15	14	ralbidva 3186	. . . . . 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 518	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
17		ralcom3 3115	. . . . 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 3186	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (∀𝑦 ∈ 𝐵 𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
20		dfss3 3926	. . 3 ⊢ (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ (topGen‘𝐶))
21		ralcom 3293	. . 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 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ⊆ wss 3905  ∪ cuni 4872  ‘cfv 6536  topGenctg 17485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-topgen 17491

This theorem is referenced by:  metss  24665  relowlssretop  38029  relowlpssretop  38030