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Theorem tgss2 20731
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 477 . . . . 5 ((𝐵𝑉 𝐵 = 𝐶) → 𝐵 = 𝐶)
2 uniexg 6920 . . . . . 6 (𝐵𝑉 𝐵 ∈ V)
32adantr 481 . . . . 5 ((𝐵𝑉 𝐵 = 𝐶) → 𝐵 ∈ V)
41, 3eqeltrrd 2699 . . . 4 ((𝐵𝑉 𝐵 = 𝐶) → 𝐶 ∈ V)
5 uniexb 6936 . . . 4 (𝐶 ∈ V ↔ 𝐶 ∈ V)
64, 5sylibr 224 . . 3 ((𝐵𝑉 𝐵 = 𝐶) → 𝐶 ∈ V)
7 tgss3 20730 . . 3 ((𝐵𝑉𝐶 ∈ V) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
86, 7syldan 487 . 2 ((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
9 eltg2b 20703 . . . . . . 7 (𝐶 ∈ V → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦)))
106, 9syl 17 . . . . . 6 ((𝐵𝑉 𝐵 = 𝐶) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦)))
11 elunii 4414 . . . . . . . . 9 ((𝑥𝑦𝑦𝐵) → 𝑥 𝐵)
1211ancoms 469 . . . . . . . 8 ((𝑦𝐵𝑥𝑦) → 𝑥 𝐵)
13 biimt 350 . . . . . . . 8 (𝑥 𝐵 → (∃𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1412, 13syl 17 . . . . . . 7 ((𝑦𝐵𝑥𝑦) → (∃𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1514ralbidva 2981 . . . . . 6 (𝑦𝐵 → (∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ ∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1610, 15sylan9bb 735 . . . . 5 (((𝐵𝑉 𝐵 = 𝐶) ∧ 𝑦𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
17 ralcom3 3099 . . . . 5 (∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)) ↔ ∀𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)))
1816, 17syl6bb 276 . . . 4 (((𝐵𝑉 𝐵 = 𝐶) ∧ 𝑦𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1918ralbidva 2981 . . 3 ((𝐵𝑉 𝐵 = 𝐶) → (∀𝑦𝐵 𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑦𝐵𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
20 dfss3 3578 . . 3 (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑦𝐵 𝑦 ∈ (topGen‘𝐶))
21 ralcom 3092 . . 3 (∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)) ↔ ∀𝑦𝐵𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)))
2219, 20, 213bitr4g 303 . 2 ((𝐵𝑉 𝐵 = 𝐶) → (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
238, 22bitrd 268 1 ((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  Vcvv 3190  wss 3560   cuni 4409  cfv 5857  topGenctg 16038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-topgen 16044
This theorem is referenced by:  metss  22253  relowlssretop  32882  relowlpssretop  32883
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