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Theorem topmeet 32484
Description: Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
topmeet ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
Distinct variable groups:   𝑗,𝑘,𝑆   𝑗,𝑉,𝑘   𝑗,𝑋,𝑘

Proof of Theorem topmeet
StepHypRef Expression
1 topmtcl 32483 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
2 inss2 3867 . . . . . . 7 (𝒫 𝑋 𝑆) ⊆ 𝑆
3 intss1 4524 . . . . . . 7 (𝑗𝑆 𝑆𝑗)
42, 3syl5ss 3647 . . . . . 6 (𝑗𝑆 → (𝒫 𝑋 𝑆) ⊆ 𝑗)
54rgen 2951 . . . . 5 𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗
6 sseq1 3659 . . . . . . 7 (𝑘 = (𝒫 𝑋 𝑆) → (𝑘𝑗 ↔ (𝒫 𝑋 𝑆) ⊆ 𝑗))
76ralbidv 3015 . . . . . 6 (𝑘 = (𝒫 𝑋 𝑆) → (∀𝑗𝑆 𝑘𝑗 ↔ ∀𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗))
87elrab 3396 . . . . 5 ((𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ↔ ((𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗))
95, 8mpbiran2 974 . . . 4 ((𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ↔ (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
101, 9sylibr 224 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
11 elssuni 4499 . . 3 ((𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} → (𝒫 𝑋 𝑆) ⊆ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
1210, 11syl 17 . 2 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ⊆ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
13 toponuni 20767 . . . . . . . . 9 (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = 𝑘)
14 eqimss2 3691 . . . . . . . . 9 (𝑋 = 𝑘 𝑘𝑋)
1513, 14syl 17 . . . . . . . 8 (𝑘 ∈ (TopOn‘𝑋) → 𝑘𝑋)
16 sspwuni 4643 . . . . . . . 8 (𝑘 ⊆ 𝒫 𝑋 𝑘𝑋)
1715, 16sylibr 224 . . . . . . 7 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
18173ad2ant2 1103 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ⊆ 𝒫 𝑋)
19 simp3 1083 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → ∀𝑗𝑆 𝑘𝑗)
20 ssint 4525 . . . . . . 7 (𝑘 𝑆 ↔ ∀𝑗𝑆 𝑘𝑗)
2119, 20sylibr 224 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 𝑆)
2218, 21ssind 3870 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ⊆ (𝒫 𝑋 𝑆))
23 selpw 4198 . . . . 5 (𝑘 ∈ 𝒫 (𝒫 𝑋 𝑆) ↔ 𝑘 ⊆ (𝒫 𝑋 𝑆))
2422, 23sylibr 224 . . . 4 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ∈ 𝒫 (𝒫 𝑋 𝑆))
2524rabssdv 3715 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ 𝒫 (𝒫 𝑋 𝑆))
26 sspwuni 4643 . . 3 ({𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ 𝒫 (𝒫 𝑋 𝑆) ↔ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ (𝒫 𝑋 𝑆))
2725, 26sylib 208 . 2 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ (𝒫 𝑋 𝑆))
2812, 27eqssd 3653 1 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468   cint 4507  cfv 5926  TopOnctopon 20763
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-int 4508  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-mre 16293  df-top 20747  df-topon 20764
This theorem is referenced by: (None)
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