Theorem tgss2 12873

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 109	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐵 = ∪ 𝐶)
2		uniexg 4424	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
3	2	adantr 274	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐵 ∈ V)
4	1, 3	eqeltrrd 2248	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐶 ∈ V)
5		uniexb 4458	. . . 4 ⊢ (𝐶 ∈ V ↔ ∪ 𝐶 ∈ V)
6	4, 5	sylibr 133	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → 𝐶 ∈ V)
7		tgss3 12872	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
8	6, 7	syldan 280	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
9		eltg2b 12848	. . . . . . 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 3801	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ ∪ 𝐵)
12	11	ancoms 266	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ ∪ 𝐵)
13		biimt 240	. . . . . . . 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 2466	. . . . . 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 459	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
17		ralcom3 2637	. . . . 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 195	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
19	18	ralbidva 2466	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (∀𝑦 ∈ 𝐵 𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
20		dfss3 3137	. . 3 ⊢ (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ (topGen‘𝐶))
21		ralcom 2633	. . 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 222	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
23	8, 22	bitrd 187	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 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  Vcvv 2730   ⊆ wss 3121  ∪ cuni 3796  ‘cfv 5198  topGenctg 12594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-topgen 12600

This theorem is referenced by:  metss  13288