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Mirrors > Home > MPE Home > Th. List > lpfval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
lpfval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
lpfval | ⊢ (𝐽 ∈ Top → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lpfval.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | topopn 21517 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
3 | pwexg 5282 | . . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V) | |
4 | mptexg 6987 | . . 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 4852 | . . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
7 | 6, 1 | syl6eqr 2877 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) |
8 | 7 | pweqd 4561 | . . . 4 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋) |
9 | fveq2 6673 | . . . . . . 7 ⊢ (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽)) | |
10 | 9 | fveq1d 6675 | . . . . . 6 ⊢ (𝑗 = 𝐽 → ((cls‘𝑗)‘(𝑥 ∖ {𝑦})) = ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))) |
11 | 10 | eleq2d 2901 | . . . . 5 ⊢ (𝑗 = 𝐽 → (𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦})) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦})))) |
12 | 11 | abbidv 2888 | . . . 4 ⊢ (𝑗 = 𝐽 → {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))} = {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) |
13 | 8, 12 | mpteq12dv 5154 | . . 3 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))})) |
14 | df-lp 21747 | . . 3 ⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))})) | |
15 | 13, 14 | fvmptg 6769 | . 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 685 | 1 ⊢ (𝐽 ∈ Top → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {cab 2802 Vcvv 3497 ∖ cdif 3936 𝒫 cpw 4542 {csn 4570 ∪ cuni 4841 ↦ cmpt 5149 ‘cfv 6358 Topctop 21504 clsccl 21629 limPtclp 21745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-top 21505 df-lp 21747 |
This theorem is referenced by: lpval 21750 |
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