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Theorem tgss2 20550
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 475 . . . . 5 ((𝐵𝑉 𝐵 = 𝐶) → 𝐵 = 𝐶)
2 uniexg 6831 . . . . . 6 (𝐵𝑉 𝐵 ∈ V)
32adantr 479 . . . . 5 ((𝐵𝑉 𝐵 = 𝐶) → 𝐵 ∈ V)
41, 3eqeltrrd 2688 . . . 4 ((𝐵𝑉 𝐵 = 𝐶) → 𝐶 ∈ V)
5 uniexb 6844 . . . 4 (𝐶 ∈ V ↔ 𝐶 ∈ V)
64, 5sylibr 222 . . 3 ((𝐵𝑉 𝐵 = 𝐶) → 𝐶 ∈ V)
7 tgss3 20549 . . 3 ((𝐵𝑉𝐶 ∈ V) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
86, 7syldan 485 . 2 ((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
9 eltg2b 20522 . . . . . . 7 (𝐶 ∈ V → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦)))
106, 9syl 17 . . . . . 6 ((𝐵𝑉 𝐵 = 𝐶) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦)))
11 elunii 4371 . . . . . . . . 9 ((𝑥𝑦𝑦𝐵) → 𝑥 𝐵)
1211ancoms 467 . . . . . . . 8 ((𝑦𝐵𝑥𝑦) → 𝑥 𝐵)
13 biimt 348 . . . . . . . 8 (𝑥 𝐵 → (∃𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1412, 13syl 17 . . . . . . 7 ((𝑦𝐵𝑥𝑦) → (∃𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1514ralbidva 2967 . . . . . 6 (𝑦𝐵 → (∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ ∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1610, 15sylan9bb 731 . . . . 5 (((𝐵𝑉 𝐵 = 𝐶) ∧ 𝑦𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
17 ralcom3 3083 . . . . 5 (∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)) ↔ ∀𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)))
1816, 17syl6bb 274 . . . 4 (((𝐵𝑉 𝐵 = 𝐶) ∧ 𝑦𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1918ralbidva 2967 . . 3 ((𝐵𝑉 𝐵 = 𝐶) → (∀𝑦𝐵 𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑦𝐵𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
20 dfss3 3557 . . 3 (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑦𝐵 𝑦 ∈ (topGen‘𝐶))
21 ralcom 3078 . . 3 (∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)) ↔ ∀𝑦𝐵𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦)))
2219, 20, 213bitr4g 301 . 2 ((𝐵𝑉 𝐵 = 𝐶) → (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
238, 22bitrd 266 1 ((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  wss 3539   cuni 4366  cfv 5790  topGenctg 15870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-topgen 15876
This theorem is referenced by:  metss  22071  relowlssretop  32211  relowlpssretop  32212
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