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Theorem tgss2 21120
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 ((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐶,𝑦,𝑧   𝑥,𝑉,𝑦
Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem tgss2
StepHypRef Expression
1 simpr 478 . . . . 5 ((𝐵𝑉 𝐵 = 𝐶) → 𝐵 = 𝐶)
2 uniexg 7189 . . . . . 6 (𝐵𝑉 𝐵 ∈ V)
32adantr 473 . . . . 5 ((𝐵𝑉 𝐵 = 𝐶) → 𝐵 ∈ V)
41, 3eqeltrrd 2879 . . . 4 ((𝐵𝑉 𝐵 = 𝐶) → 𝐶 ∈ V)
5 uniexb 7206 . . . 4 (𝐶 ∈ V ↔ 𝐶 ∈ V)
64, 5sylibr 226 . . 3 ((𝐵𝑉 𝐵 = 𝐶) → 𝐶 ∈ V)
7 tgss3 21119 . . 3 ((𝐵𝑉𝐶 ∈ V) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
86, 7syldan 586 . 2 ((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
9 eltg2b 21092 . . . . . . 7 (𝐶 ∈ V → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦)))
106, 9syl 17 . . . . . 6 ((𝐵𝑉 𝐵 = 𝐶) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦)))
11 elunii 4633 . . . . . . . . 9 ((𝑥𝑦𝑦𝐵) → 𝑥 𝐵)
1211ancoms 451 . . . . . . . 8 ((𝑦𝐵𝑥𝑦) → 𝑥 𝐵)
13 biimt 352 . . . . . . . 8 (𝑥 𝐵 → (∃𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1412, 13syl 17 . . . . . . 7 ((𝑦𝐵𝑥𝑦) → (∃𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1514ralbidva 3166 . . . . . 6 (𝑦𝐵 → (∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ ∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1610, 15sylan9bb 506 . . . . 5 (((𝐵𝑉 𝐵 = 𝐶) ∧ 𝑦𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
17 ralcom3 3286 . . . . 5 (∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)) ↔ ∀𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)))
1816, 17syl6bb 279 . . . 4 (((𝐵𝑉 𝐵 = 𝐶) ∧ 𝑦𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1918ralbidva 3166 . . 3 ((𝐵𝑉 𝐵 = 𝐶) → (∀𝑦𝐵 𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑦𝐵𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
20 dfss3 3787 . . 3 (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑦𝐵 𝑦 ∈ (topGen‘𝐶))
21 ralcom 3279 . . 3 (∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)) ↔ ∀𝑦𝐵𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)))
2219, 20, 213bitr4g 306 . 2 ((𝐵𝑉 𝐵 = 𝐶) → (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
238, 22bitrd 271 1 ((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  wrex 3090  Vcvv 3385  wss 3769   cuni 4628  cfv 6101  topGenctg 16413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-topgen 16419
This theorem is referenced by:  metss  22641  relowlssretop  33709  relowlpssretop  33710
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