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Theorem topmtcl 34198
Description: The meet of a collection of topologies on 𝑋 is again a topology on 𝑋. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
topmtcl ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))

Proof of Theorem topmtcl
StepHypRef Expression
1 toponmre 21847 . 2 (𝑋𝑉 → (TopOn‘𝑋) ∈ (Moore‘𝒫 𝑋))
2 mrerintcl 16974 . 2 (((TopOn‘𝑋) ∈ (Moore‘𝒫 𝑋) ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
31, 2sylan 583 1 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2114  cin 3843  wss 3844  𝒫 cpw 4489   cint 4837  cfv 6340  Moorecmre 16959  TopOnctopon 21664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-int 4838  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-iota 6298  df-fun 6342  df-fv 6348  df-mre 16963  df-top 21648  df-topon 21665
This theorem is referenced by:  topmeet  34199
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