Theorem tgss2 12248

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 4361	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
3	2	adantr 274	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐵 ∈ V)
4	1, 3	eqeltrrd 2217	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ∪ 𝐶 ∈ V)
5		uniexb 4394	. . . 4 ⊢ (𝐶 ∈ V ↔ ∪ 𝐶 ∈ V)
6	4, 5	sylibr 133	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → 𝐶 ∈ V)
7		tgss3 12247	. . 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 12223	. . . . . . 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 3741	. . . . . . . . 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 2433	. . . . . 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 457	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ 𝑦 (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
17		ralcom3 2598	. . . . 5 ⊢ (∀𝑥 ∈ 𝑦 (𝑥 ∈ ∪ 𝐵 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
18	16, 17	syl6bb 195	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
19	18	ralbidva 2433	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → (∀𝑦 ∈ 𝐵 𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ ∪ 𝐵(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
20		dfss3 3087	. . 3 ⊢ (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ (topGen‘𝐶))
21		ralcom 2594	. . 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 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  Vcvv 2686   ⊆ wss 3071  ∪ cuni 3736  ‘cfv 5123  topGenctg 12135

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-topgen 12141

This theorem is referenced by:  metss  12663