Theorem clsf 23081

Description: The closure function is a function from subsets of the base to closed sets. (Contributed by Mario Carneiro, 11-Apr-2015.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clsf	⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))

Proof of Theorem clsf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4556	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
2		clscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	clsval 23070	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) = ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦})
4		fvex 6869	. . . 4 ⊢ ((cls‘𝐽)‘𝑥) ∈ V
5	3, 4	eqeltrrdi 2865	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦} ∈ V)
6	1, 5	sylan2 601	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 𝑋) → ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦} ∈ V)
7	2	clsfval 23058	. 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}))
8		elpwi 4556	. . 3 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9	2	clscld 23080	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
10	8, 9	sylan2 601	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
11	6, 7, 10	fmpt2d 7095	1 ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1554   ∈ wcel 2136  {crab 3408  Vcvv 3448   ⊆ wss 3899  𝒫 cpw 4549  ∪ cuni 4859  ∩ cint 4899  ⟶wf 6506  ‘cfv 6510  Topctop 22926  Clsdccld 23049  clsccl 23051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4900  df-iun 4945  df-iin 4946  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-top 22927  df-cld 23052  df-cls 23054

This theorem is referenced by:  clsf2  44650