MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgss2 Structured version   Visualization version   GIF version

Theorem tgss2 20839
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 476 . . . . 5 ((𝐵𝑉 𝐵 = 𝐶) → 𝐵 = 𝐶)
2 uniexg 6997 . . . . . 6 (𝐵𝑉 𝐵 ∈ V)
32adantr 480 . . . . 5 ((𝐵𝑉 𝐵 = 𝐶) → 𝐵 ∈ V)
41, 3eqeltrrd 2731 . . . 4 ((𝐵𝑉 𝐵 = 𝐶) → 𝐶 ∈ V)
5 uniexb 7015 . . . 4 (𝐶 ∈ V ↔ 𝐶 ∈ V)
64, 5sylibr 224 . . 3 ((𝐵𝑉 𝐵 = 𝐶) → 𝐶 ∈ V)
7 tgss3 20838 . . 3 ((𝐵𝑉𝐶 ∈ V) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
86, 7syldan 486 . 2 ((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
9 eltg2b 20811 . . . . . . 7 (𝐶 ∈ V → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦)))
106, 9syl 17 . . . . . 6 ((𝐵𝑉 𝐵 = 𝐶) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦)))
11 elunii 4473 . . . . . . . . 9 ((𝑥𝑦𝑦𝐵) → 𝑥 𝐵)
1211ancoms 468 . . . . . . . 8 ((𝑦𝐵𝑥𝑦) → 𝑥 𝐵)
13 biimt 349 . . . . . . . 8 (𝑥 𝐵 → (∃𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1412, 13syl 17 . . . . . . 7 ((𝑦𝐵𝑥𝑦) → (∃𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1514ralbidva 3014 . . . . . 6 (𝑦𝐵 → (∀𝑥𝑦𝑧𝐶 (𝑥𝑧𝑧𝑦) ↔ ∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
1610, 15sylan9bb 736 . . . . 5 (((𝐵𝑉 𝐵 = 𝐶) ∧ 𝑦𝐵) → (𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑥𝑦 (𝑥 𝐵 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
17 ralcom3 3134 . . . . 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 3014 . . 3 ((𝐵𝑉 𝐵 = 𝐶) → (∀𝑦𝐵 𝑦 ∈ (topGen‘𝐶) ↔ ∀𝑦𝐵𝑥 𝐵(𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
20 dfss3 3625 . . 3 (𝐵 ⊆ (topGen‘𝐶) ↔ ∀𝑦𝐵 𝑦 ∈ (topGen‘𝐶))
21 ralcom 3127 . . 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 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607   cuni 4468  cfv 5926  topGenctg 16145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151
This theorem is referenced by:  metss  22360  relowlssretop  33341  relowlpssretop  33342
  Copyright terms: Public domain W3C validator