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Theorem topmeet 33786
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 33785 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
2 inss2 4180 . . . . . . 7 (𝒫 𝑋 𝑆) ⊆ 𝑆
3 intss1 4866 . . . . . . 7 (𝑗𝑆 𝑆𝑗)
42, 3sstrid 3953 . . . . . 6 (𝑗𝑆 → (𝒫 𝑋 𝑆) ⊆ 𝑗)
54rgen 3140 . . . . 5 𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗
6 sseq1 3967 . . . . . . 7 (𝑘 = (𝒫 𝑋 𝑆) → (𝑘𝑗 ↔ (𝒫 𝑋 𝑆) ⊆ 𝑗))
76ralbidv 3187 . . . . . 6 (𝑘 = (𝒫 𝑋 𝑆) → (∀𝑗𝑆 𝑘𝑗 ↔ ∀𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗))
87elrab 3655 . . . . 5 ((𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ↔ ((𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗))
95, 8mpbiran2 709 . . . 4 ((𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ↔ (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
101, 9sylibr 237 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
11 elssuni 4843 . . 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 21517 . . . . . . . . 9 (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = 𝑘)
14 eqimss2 3999 . . . . . . . . 9 (𝑋 = 𝑘 𝑘𝑋)
1513, 14syl 17 . . . . . . . 8 (𝑘 ∈ (TopOn‘𝑋) → 𝑘𝑋)
16 sspwuni 4997 . . . . . . . 8 (𝑘 ⊆ 𝒫 𝑋 𝑘𝑋)
1715, 16sylibr 237 . . . . . . 7 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
18173ad2ant2 1131 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ⊆ 𝒫 𝑋)
19 simp3 1135 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → ∀𝑗𝑆 𝑘𝑗)
20 ssint 4867 . . . . . . 7 (𝑘 𝑆 ↔ ∀𝑗𝑆 𝑘𝑗)
2119, 20sylibr 237 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 𝑆)
2218, 21ssind 4183 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ⊆ (𝒫 𝑋 𝑆))
23 velpw 4516 . . . . 5 (𝑘 ∈ 𝒫 (𝒫 𝑋 𝑆) ↔ 𝑘 ⊆ (𝒫 𝑋 𝑆))
2422, 23sylibr 237 . . . 4 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ∈ 𝒫 (𝒫 𝑋 𝑆))
2524rabssdv 4026 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ 𝒫 (𝒫 𝑋 𝑆))
26 sspwuni 4997 . . 3 ({𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ 𝒫 (𝒫 𝑋 𝑆) ↔ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ (𝒫 𝑋 𝑆))
2725, 26sylib 221 . 2 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ (𝒫 𝑋 𝑆))
2812, 27eqssd 3959 1 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  {crab 3134  cin 3907  wss 3908  𝒫 cpw 4511   cuni 4813   cint 4851  cfv 6334  TopOnctopon 21513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-mre 16848  df-top 21497  df-topon 21514
This theorem is referenced by: (None)
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