Theorem topmtcl 34479

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 22152	. 2 ⊢ (𝑋 ∈ 𝑉 → (TopOn‘𝑋) ∈ (Moore‘𝒫 𝑋))
2		mrerintcl 17223	. 2 ⊢ (((TopOn‘𝑋) ∈ (Moore‘𝒫 𝑋) ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))
3	1, 2	sylan 579	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∩ cint 4876  ‘cfv 6418  Moorecmre 17208  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:  topmeet  34480