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Theorem cldllycmp 23219
Description: A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 23210.) (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
cldllycmp ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ (𝐽 β†Ύt 𝐴) ∈ 𝑛-Locally Comp)
Proof of Theorem cldllycmp
Dummy variables 𝑒 𝑣 𝑀 π‘₯ 𝑦 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nllytop 23197 . . 3 (𝐽 ∈ 𝑛-Locally Comp β†’ 𝐽 ∈ Top)
2 resttop 22884 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ (𝐽 β†Ύt 𝐴) ∈ Top)
31, 2sylan 578 . 2 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ (𝐽 β†Ύt 𝐴) ∈ Top)
4 elrest 17377 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ (π‘₯ ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ 𝐴)))
5 simpll 763 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ 𝐽 ∈ 𝑛-Locally Comp)
6 simprl 767 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ 𝑒 ∈ 𝐽)
7 simprr 769 . . . . . . . . . . 11 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ 𝑦 ∈ (𝑒 ∩ 𝐴))
87elin1d 4197 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ 𝑦 ∈ 𝑒)
9 nlly2i 23200 . . . . . . . . . 10 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑒 ∈ 𝐽 ∧ 𝑦 ∈ 𝑒) β†’ βˆƒπ‘  ∈ 𝒫 π‘’βˆƒπ‘€ ∈ 𝐽 (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))
105, 6, 8, 9syl3anc 1369 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ βˆƒπ‘  ∈ 𝒫 π‘’βˆƒπ‘€ ∈ 𝐽 (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))
113ad2antrr 722 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝐽 β†Ύt 𝐴) ∈ Top)
121ad3antrrr 726 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝐽 ∈ Top)
13 simpllr 772 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝐴 ∈ (Clsdβ€˜π½))
14 simprlr 776 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑀 ∈ 𝐽)
15 elrestr 17378 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsdβ€˜π½) ∧ 𝑀 ∈ 𝐽) β†’ (𝑀 ∩ 𝐴) ∈ (𝐽 β†Ύt 𝐴))
1612, 13, 14, 15syl3anc 1369 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑀 ∩ 𝐴) ∈ (𝐽 β†Ύt 𝐴))
17 simprr1 1219 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑦 ∈ 𝑀)
18 simplrr 774 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑦 ∈ (𝑒 ∩ 𝐴))
1918elin2d 4198 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑦 ∈ 𝐴)
2017, 19elind 4193 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑦 ∈ (𝑀 ∩ 𝐴))
21 opnneip 22843 . . . . . . . . . . . . . . 15 (((𝐽 β†Ύt 𝐴) ∈ Top ∧ (𝑀 ∩ 𝐴) ∈ (𝐽 β†Ύt 𝐴) ∧ 𝑦 ∈ (𝑀 ∩ 𝐴)) β†’ (𝑀 ∩ 𝐴) ∈ ((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}))
2211, 16, 20, 21syl3anc 1369 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑀 ∩ 𝐴) ∈ ((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}))
23 simprr2 1220 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑀 βŠ† 𝑠)
2423ssrind 4234 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑀 ∩ 𝐴) βŠ† (𝑠 ∩ 𝐴))
25 inss2 4228 . . . . . . . . . . . . . . 15 (𝑠 ∩ 𝐴) βŠ† 𝐴
26 eqid 2730 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝐽 = βˆͺ 𝐽
2726cldss 22753 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (Clsdβ€˜π½) β†’ 𝐴 βŠ† βˆͺ 𝐽)
2813, 27syl 17 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝐴 βŠ† βˆͺ 𝐽)
2926restuni 22886 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴 βŠ† βˆͺ 𝐽) β†’ 𝐴 = βˆͺ (𝐽 β†Ύt 𝐴))
3012, 28, 29syl2anc 582 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝐴 = βˆͺ (𝐽 β†Ύt 𝐴))
3125, 30sseqtrid 4033 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑠 ∩ 𝐴) βŠ† βˆͺ (𝐽 β†Ύt 𝐴))
32 eqid 2730 . . . . . . . . . . . . . . 15 βˆͺ (𝐽 β†Ύt 𝐴) = βˆͺ (𝐽 β†Ύt 𝐴)
3332ssnei2 22840 . . . . . . . . . . . . . 14 ((((𝐽 β†Ύt 𝐴) ∈ Top ∧ (𝑀 ∩ 𝐴) ∈ ((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦})) ∧ ((𝑀 ∩ 𝐴) βŠ† (𝑠 ∩ 𝐴) ∧ (𝑠 ∩ 𝐴) βŠ† βˆͺ (𝐽 β†Ύt 𝐴))) β†’ (𝑠 ∩ 𝐴) ∈ ((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}))
3411, 22, 24, 31, 33syl22anc 835 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑠 ∩ 𝐴) ∈ ((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}))
35 simprll 775 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑠 ∈ 𝒫 𝑒)
3635elpwid 4610 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑠 βŠ† 𝑒)
3736ssrind 4234 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑠 ∩ 𝐴) βŠ† (𝑒 ∩ 𝐴))
38 vex 3476 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
3938inex1 5316 . . . . . . . . . . . . . . 15 (𝑠 ∩ 𝐴) ∈ V
4039elpw 4605 . . . . . . . . . . . . . 14 ((𝑠 ∩ 𝐴) ∈ 𝒫 (𝑒 ∩ 𝐴) ↔ (𝑠 ∩ 𝐴) βŠ† (𝑒 ∩ 𝐴))
4137, 40sylibr 233 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑠 ∩ 𝐴) ∈ 𝒫 (𝑒 ∩ 𝐴))
4234, 41elind 4193 . . . . . . . . . . . 12 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑠 ∩ 𝐴) ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴)))
4325a1i 11 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑠 ∩ 𝐴) βŠ† 𝐴)
44 restabs 22889 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (𝑠 ∩ 𝐴) βŠ† 𝐴 ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ ((𝐽 β†Ύt 𝐴) β†Ύt (𝑠 ∩ 𝐴)) = (𝐽 β†Ύt (𝑠 ∩ 𝐴)))
4512, 43, 13, 44syl3anc 1369 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ ((𝐽 β†Ύt 𝐴) β†Ύt (𝑠 ∩ 𝐴)) = (𝐽 β†Ύt (𝑠 ∩ 𝐴)))
46 inss1 4227 . . . . . . . . . . . . . . . 16 (𝑠 ∩ 𝐴) βŠ† 𝑠
4746a1i 11 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑠 ∩ 𝐴) βŠ† 𝑠)
48 restabs 22889 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (𝑠 ∩ 𝐴) βŠ† 𝑠 ∧ 𝑠 ∈ 𝒫 𝑒) β†’ ((𝐽 β†Ύt 𝑠) β†Ύt (𝑠 ∩ 𝐴)) = (𝐽 β†Ύt (𝑠 ∩ 𝐴)))
4912, 47, 35, 48syl3anc 1369 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ ((𝐽 β†Ύt 𝑠) β†Ύt (𝑠 ∩ 𝐴)) = (𝐽 β†Ύt (𝑠 ∩ 𝐴)))
5045, 49eqtr4d 2773 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ ((𝐽 β†Ύt 𝐴) β†Ύt (𝑠 ∩ 𝐴)) = ((𝐽 β†Ύt 𝑠) β†Ύt (𝑠 ∩ 𝐴)))
51 simprr3 1221 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝐽 β†Ύt 𝑠) ∈ Comp)
52 incom 4200 . . . . . . . . . . . . . . 15 (𝑠 ∩ 𝐴) = (𝐴 ∩ 𝑠)
53 eqid 2730 . . . . . . . . . . . . . . . . 17 (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)
54 ineq1 4204 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 β†’ (𝑣 ∩ 𝑠) = (𝐴 ∩ 𝑠))
5554rspceeqv 3632 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (Clsdβ€˜π½) ∧ (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)) β†’ βˆƒπ‘£ ∈ (Clsdβ€˜π½)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠))
5613, 53, 55sylancl 584 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ βˆƒπ‘£ ∈ (Clsdβ€˜π½)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠))
57 simplrl 773 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑒 ∈ 𝐽)
58 elssuni 4940 . . . . . . . . . . . . . . . . . . 19 (𝑒 ∈ 𝐽 β†’ 𝑒 βŠ† βˆͺ 𝐽)
5957, 58syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑒 βŠ† βˆͺ 𝐽)
6036, 59sstrd 3991 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ 𝑠 βŠ† βˆͺ 𝐽)
6126restcld 22896 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑠 βŠ† βˆͺ 𝐽) β†’ ((𝐴 ∩ 𝑠) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝑠)) ↔ βˆƒπ‘£ ∈ (Clsdβ€˜π½)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠)))
6212, 60, 61syl2anc 582 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ ((𝐴 ∩ 𝑠) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝑠)) ↔ βˆƒπ‘£ ∈ (Clsdβ€˜π½)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠)))
6356, 62mpbird 256 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝐴 ∩ 𝑠) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝑠)))
6452, 63eqeltrid 2835 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ (𝑠 ∩ 𝐴) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝑠)))
65 cmpcld 23126 . . . . . . . . . . . . . 14 (((𝐽 β†Ύt 𝑠) ∈ Comp ∧ (𝑠 ∩ 𝐴) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝑠))) β†’ ((𝐽 β†Ύt 𝑠) β†Ύt (𝑠 ∩ 𝐴)) ∈ Comp)
6651, 64, 65syl2anc 582 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ ((𝐽 β†Ύt 𝑠) β†Ύt (𝑠 ∩ 𝐴)) ∈ Comp)
6750, 66eqeltrd 2831 . . . . . . . . . . . 12 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ ((𝐽 β†Ύt 𝐴) β†Ύt (𝑠 ∩ 𝐴)) ∈ Comp)
68 oveq2 7419 . . . . . . . . . . . . . 14 (𝑣 = (𝑠 ∩ 𝐴) β†’ ((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) = ((𝐽 β†Ύt 𝐴) β†Ύt (𝑠 ∩ 𝐴)))
6968eleq1d 2816 . . . . . . . . . . . . 13 (𝑣 = (𝑠 ∩ 𝐴) β†’ (((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp ↔ ((𝐽 β†Ύt 𝐴) β†Ύt (𝑠 ∩ 𝐴)) ∈ Comp))
7069rspcev 3611 . . . . . . . . . . . 12 (((𝑠 ∩ 𝐴) ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴)) ∧ ((𝐽 β†Ύt 𝐴) β†Ύt (𝑠 ∩ 𝐴)) ∈ Comp) β†’ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴))((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp)
7142, 67, 70syl2anc 582 . . . . . . . . . . 11 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp))) β†’ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴))((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp)
7271expr 455 . . . . . . . . . 10 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) ∧ (𝑠 ∈ 𝒫 𝑒 ∧ 𝑀 ∈ 𝐽)) β†’ ((𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp) β†’ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴))((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp))
7372rexlimdvva 3209 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ (βˆƒπ‘  ∈ 𝒫 π‘’βˆƒπ‘€ ∈ 𝐽 (𝑦 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑠 ∧ (𝐽 β†Ύt 𝑠) ∈ Comp) β†’ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴))((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp))
7410, 73mpd 15 . . . . . . . 8 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴))((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp)
7574anassrs 466 . . . . . . 7 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑦 ∈ (𝑒 ∩ 𝐴)) β†’ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴))((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp)
7675ralrimiva 3144 . . . . . 6 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ 𝑒 ∈ 𝐽) β†’ βˆ€π‘¦ ∈ (𝑒 ∩ 𝐴)βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴))((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp)
77 pweq 4615 . . . . . . . . 9 (π‘₯ = (𝑒 ∩ 𝐴) β†’ 𝒫 π‘₯ = 𝒫 (𝑒 ∩ 𝐴))
7877ineq2d 4211 . . . . . . . 8 (π‘₯ = (𝑒 ∩ 𝐴) β†’ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 π‘₯) = (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴)))
7978rexeqdv 3324 . . . . . . 7 (π‘₯ = (𝑒 ∩ 𝐴) β†’ (βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 π‘₯)((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp ↔ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴))((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp))
8079raleqbi1dv 3331 . . . . . 6 (π‘₯ = (𝑒 ∩ 𝐴) β†’ (βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 π‘₯)((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp ↔ βˆ€π‘¦ ∈ (𝑒 ∩ 𝐴)βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 (𝑒 ∩ 𝐴))((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp))
8176, 80syl5ibrcom 246 . . . . 5 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ 𝑒 ∈ 𝐽) β†’ (π‘₯ = (𝑒 ∩ 𝐴) β†’ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 π‘₯)((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp))
8281rexlimdva 3153 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ (βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ 𝐴) β†’ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 π‘₯)((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp))
834, 82sylbid 239 . . 3 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ (π‘₯ ∈ (𝐽 β†Ύt 𝐴) β†’ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 π‘₯)((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp))
8483ralrimiv 3143 . 2 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ βˆ€π‘₯ ∈ (𝐽 β†Ύt 𝐴)βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 π‘₯)((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp)
85 isnlly 23193 . 2 ((𝐽 β†Ύt 𝐴) ∈ 𝑛-Locally Comp ↔ ((𝐽 β†Ύt 𝐴) ∈ Top ∧ βˆ€π‘₯ ∈ (𝐽 β†Ύt 𝐴)βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘£ ∈ (((neiβ€˜(𝐽 β†Ύt 𝐴))β€˜{𝑦}) ∩ 𝒫 π‘₯)((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp))
863, 84, 85sylanbrc 581 1 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ (𝐽 β†Ύt 𝐴) ∈ 𝑛-Locally Comp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907  β€˜cfv 6542  (class class class)co 7411   β†Ύt crest 17370  Topctop 22615  Clsdccld 22740  neicnei 22821  Compccmp 23110  π‘›-Locally cnlly 23189
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-fin 8945  df-fi 9408  df-rest 17372  df-topgen 17393  df-top 22616  df-topon 22633  df-bases 22669  df-cld 22743  df-nei 22822  df-cmp 23111  df-nlly 23191
This theorem is referenced by:  rellycmp  24703
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