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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 ((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 488 . . . . 5 ((𝐵𝑉 𝐵 = 𝐶) → 𝐵 = 𝐶)
2 uniexg 7478 . . . . . 6 (𝐵𝑉 𝐵 ∈ V)
32adantr 484 . . . . 5 ((𝐵𝑉 𝐵 = 𝐶) → 𝐵 ∈ V)
41, 3eqeltrrd 2834 . . . 4 ((𝐵𝑉 𝐵 = 𝐶) → 𝐶 ∈ V)
5 uniexb 7499 . . . 4 (𝐶 ∈ V ↔ 𝐶 ∈ V)
64, 5sylibr 237 . . 3 ((𝐵𝑉 𝐵 = 𝐶) → 𝐶 ∈ V)
7 tgss3 21730 . . 3 ((𝐵𝑉𝐶 ∈ V) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
86, 7syldan 594 . 2 ((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
9 eltg2b 21703 . . . . . . 7 (𝐶 ∈ V → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦)))
106, 9syl 17 . . . . . 6 ((𝐵𝑉 𝐵 = 𝐶) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦)))
11 elunii 4798 . . . . . . . . 9 ((𝑥𝑦𝑦𝐵) → 𝑥 𝐵)
1211ancoms 462 . . . . . . . 8 ((𝑦𝐵𝑥𝑦) → 𝑥 𝐵)
13 biimt 364 . . . . . . . 8 (𝑥 𝐵 → (∃𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1412, 13syl 17 . . . . . . 7 ((𝑦𝐵𝑥𝑦) → (∃𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1514ralbidva 3108 . . . . . 6 (𝑦𝐵 → (∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ ∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1610, 15sylan9bb 513 . . . . 5 (((𝐵𝑉 𝐵 = 𝐶) ∧ 𝑦𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
17 ralcom3 3266 . . . . 5 (∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)) ↔ ∀𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)))
1816, 17bitrdi 290 . . . 4 (((𝐵𝑉 𝐵 = 𝐶) ∧ 𝑦𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1918ralbidva 3108 . . 3 ((𝐵𝑉 𝐵 = 𝐶) → (∀𝑦𝐵 𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑦𝐵𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
20 dfss3 3863 . . 3 (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑦𝐵 𝑦 ∈ (topGen‘𝐶))
21 ralcom 3257 . . 3 (∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)) ↔ ∀𝑦𝐵𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)))
2219, 20, 213bitr4g 317 . 2 ((𝐵𝑉 𝐵 = 𝐶) → (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
238, 22bitrd 282 1 ((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
Colors of variables: wff setvar class
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
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