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Theorem topmeet 34480
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 34479 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
2 inss2 4160 . . . . . . 7 (𝒫 𝑋 𝑆) ⊆ 𝑆
3 intss1 4891 . . . . . . 7 (𝑗𝑆 𝑆𝑗)
42, 3sstrid 3928 . . . . . 6 (𝑗𝑆 → (𝒫 𝑋 𝑆) ⊆ 𝑗)
54rgen 3073 . . . . 5 𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗
6 sseq1 3942 . . . . . . 7 (𝑘 = (𝒫 𝑋 𝑆) → (𝑘𝑗 ↔ (𝒫 𝑋 𝑆) ⊆ 𝑗))
76ralbidv 3120 . . . . . 6 (𝑘 = (𝒫 𝑋 𝑆) → (∀𝑗𝑆 𝑘𝑗 ↔ ∀𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗))
87elrab 3617 . . . . 5 ((𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ↔ ((𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗))
95, 8mpbiran2 706 . . . 4 ((𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ↔ (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
101, 9sylibr 233 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
11 elssuni 4868 . . 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 21971 . . . . . . . . 9 (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = 𝑘)
14 eqimss2 3974 . . . . . . . . 9 (𝑋 = 𝑘 𝑘𝑋)
1513, 14syl 17 . . . . . . . 8 (𝑘 ∈ (TopOn‘𝑋) → 𝑘𝑋)
16 sspwuni 5025 . . . . . . . 8 (𝑘 ⊆ 𝒫 𝑋 𝑘𝑋)
1715, 16sylibr 233 . . . . . . 7 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
18173ad2ant2 1132 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ⊆ 𝒫 𝑋)
19 simp3 1136 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → ∀𝑗𝑆 𝑘𝑗)
20 ssint 4892 . . . . . . 7 (𝑘 𝑆 ↔ ∀𝑗𝑆 𝑘𝑗)
2119, 20sylibr 233 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 𝑆)
2218, 21ssind 4163 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ⊆ (𝒫 𝑋 𝑆))
23 velpw 4535 . . . . 5 (𝑘 ∈ 𝒫 (𝒫 𝑋 𝑆) ↔ 𝑘 ⊆ (𝒫 𝑋 𝑆))
2422, 23sylibr 233 . . . 4 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ∈ 𝒫 (𝒫 𝑋 𝑆))
2524rabssdv 4004 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ 𝒫 (𝒫 𝑋 𝑆))
26 sspwuni 5025 . . 3 ({𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ 𝒫 (𝒫 𝑋 𝑆) ↔ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ (𝒫 𝑋 𝑆))
2725, 26sylib 217 . 2 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ (𝒫 𝑋 𝑆))
2812, 27eqssd 3934 1 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1539  wcel 2108  wral 3063  {crab 3067  cin 3882  wss 3883  𝒫 cpw 4530   cuni 4836   cint 4876  cfv 6418  TopOnctopon 21967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-mre 17212  df-top 21951  df-topon 21968
This theorem is referenced by: (None)
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