Theorem topmtcl 36364

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	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))

Proof of Theorem topmtcl

Step	Hyp	Ref	Expression
1		toponmre 23101	. 2 ⊢ (𝑋 ∈ 𝑉 → (TopOn‘𝑋) ∈ (Moore‘𝒫 𝑋))
2		mrerintcl 17640	. 2 ⊢ (((TopOn‘𝑋) ∈ (Moore‘𝒫 𝑋) ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))
3	1, 2	sylan 580	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ∩ cint 4946  ‘cfv 6561  Moorecmre 17625  TopOnctopon 22916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-mre 17629  df-top 22900  df-topon 22917

This theorem is referenced by:  topmeet  36365