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Theorem topmeet 31335
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 31334 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
2 inss2 3795 . . . . . . 7 (𝒫 𝑋 𝑆) ⊆ 𝑆
3 intss1 4421 . . . . . . 7 (𝑗𝑆 𝑆𝑗)
42, 3syl5ss 3578 . . . . . 6 (𝑗𝑆 → (𝒫 𝑋 𝑆) ⊆ 𝑗)
54rgen 2905 . . . . 5 𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗
6 sseq1 3588 . . . . . . 7 (𝑘 = (𝒫 𝑋 𝑆) → (𝑘𝑗 ↔ (𝒫 𝑋 𝑆) ⊆ 𝑗))
76ralbidv 2968 . . . . . 6 (𝑘 = (𝒫 𝑋 𝑆) → (∀𝑗𝑆 𝑘𝑗 ↔ ∀𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗))
87elrab 3330 . . . . 5 ((𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ↔ ((𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 (𝒫 𝑋 𝑆) ⊆ 𝑗))
95, 8mpbiran2 955 . . . 4 ((𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ↔ (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
101, 9sylibr 222 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
11 elssuni 4397 . . 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 20484 . . . . . . . . 9 (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = 𝑘)
14 eqimss2 3620 . . . . . . . . 9 (𝑋 = 𝑘 𝑘𝑋)
1513, 14syl 17 . . . . . . . 8 (𝑘 ∈ (TopOn‘𝑋) → 𝑘𝑋)
16 sspwuni 4541 . . . . . . . 8 (𝑘 ⊆ 𝒫 𝑋 𝑘𝑋)
1715, 16sylibr 222 . . . . . . 7 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
18173ad2ant2 1075 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ⊆ 𝒫 𝑋)
19 simp3 1055 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → ∀𝑗𝑆 𝑘𝑗)
20 ssint 4422 . . . . . . 7 (𝑘 𝑆 ↔ ∀𝑗𝑆 𝑘𝑗)
2119, 20sylibr 222 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 𝑆)
2218, 21ssind 3798 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ⊆ (𝒫 𝑋 𝑆))
23 selpw 4114 . . . . 5 (𝑘 ∈ 𝒫 (𝒫 𝑋 𝑆) ↔ 𝑘 ⊆ (𝒫 𝑋 𝑆))
2422, 23sylibr 222 . . . 4 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑘𝑗) → 𝑘 ∈ 𝒫 (𝒫 𝑋 𝑆))
2524rabssdv 3644 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ 𝒫 (𝒫 𝑋 𝑆))
26 sspwuni 4541 . . 3 ({𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ 𝒫 (𝒫 𝑋 𝑆) ↔ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ (𝒫 𝑋 𝑆))
2725, 26sylib 206 . 2 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗} ⊆ (𝒫 𝑋 𝑆))
2812, 27eqssd 3584 1 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  {crab 2899  cin 3538  wss 3539  𝒫 cpw 4107   cuni 4366   cint 4404  cfv 5790  TopOnctopon 20460
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 6824
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-int 4405  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-mre 16015  df-top 20463  df-topon 20465
This theorem is referenced by: (None)
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