Theorem lpfval 23113

Description: The limit point function on the subsets of a topology's base set. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
lpfval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
lpfval	⊢ (𝐽 ∈ Top → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦

Proof of Theorem lpfval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpfval.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	topopn 22881	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3		pwexg 5315	. . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
4		mptexg 7169	. . 3 ⊢ (𝒫 𝑋 ∈ V → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) ∈ V)
5	2, 3, 4	3syl 18	. 2 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) ∈ V)
6		unieq 4862	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
7	6, 1	eqtr4di 2790	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
8	7	pweqd 4559	. . . 4 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
9		fveq2 6834	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
10	9	fveq1d 6836	. . . . . 6 ⊢ (𝑗 = 𝐽 → ((cls‘𝑗)‘(𝑥 ∖ {𝑦})) = ((cls‘𝐽)‘(𝑥 ∖ {𝑦})))
11	10	eleq2d 2823	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦})) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))))
12	11	abbidv 2803	. . . 4 ⊢ (𝑗 = 𝐽 → {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))} = {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))})
13	8, 12	mpteq12dv 5173	. . 3 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
14		df-lp 23111	. . 3 ⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
15	13, 14	fvmptg 6939	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) ∈ V) → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
16	5, 15	mpdan 688	1 ⊢ (𝐽 ∈ Top → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {cab 2715  Vcvv 3430   ∖ cdif 3887  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851   ↦ cmpt 5167  ‘cfv 6492  Topctop 22868  clsccl 22993  limPtclp 23109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22869  df-lp 23111

This theorem is referenced by:  lpval  23114