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Theorem lpfval 21221
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 ↦ {𝑦𝑦 ∈ ((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.
StepHypRef Expression
1 lpfval.1 . . . 4 𝑋 = 𝐽
21topopn 20989 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 5013 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 6676 . . 3 (𝒫 𝑋 ∈ V → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) ∈ V)
52, 3, 43syl 18 . 2 (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) ∈ V)
6 unieq 4601 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1syl6eqr 2816 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4319 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 fveq2 6374 . . . . . . 7 (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
109fveq1d 6376 . . . . . 6 (𝑗 = 𝐽 → ((cls‘𝑗)‘(𝑥 ∖ {𝑦})) = ((cls‘𝐽)‘(𝑥 ∖ {𝑦})))
1110eleq2d 2829 . . . . 5 (𝑗 = 𝐽 → (𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦})) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))))
1211abbidv 2883 . . . 4 (𝑗 = 𝐽 → {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))} = {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))})
138, 12mpteq12dv 4891 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
14 df-lp 21219 . . 3 limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
1513, 14fvmptg 6468 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) ∈ V) → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
165, 15mpdan 678 1 (𝐽 ∈ Top → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  {cab 2750  Vcvv 3349  cdif 3728  𝒫 cpw 4314  {csn 4333   cuni 4593  cmpt 4887  cfv 6067  Topctop 20976  clsccl 21101  limPtclp 21217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-top 20977  df-lp 21219
This theorem is referenced by:  lpval  21222
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