Theorem clsf 21650

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 4550	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
2		clscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	clsval 21639	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) = ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦})
4		fvex 6677	. . . 4 ⊢ ((cls‘𝐽)‘𝑥) ∈ V
5	3, 4	eqeltrrdi 2922	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦} ∈ V)
6	1, 5	sylan2 594	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 𝑋) → ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦} ∈ V)
7	2	clsfval 21627	. 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}))
8		elpwi 4550	. . 3 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9	2	clscld 21649	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
10	8, 9	sylan2 594	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
11	6, 7, 10	fmpt2d 6881	1 ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {crab 3142  Vcvv 3494   ⊆ wss 3935  𝒫 cpw 4538  ∪ cuni 4831  ∩ cint 4868  ⟶wf 6345  ‘cfv 6349  Topctop 21495  Clsdccld 21618  clsccl 21620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-top 21496  df-cld 21621  df-cls 21623

This theorem is referenced by:  clsf2  40469