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Theorem cldllycmp 22999
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 22990.) (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 22977 . . 3 (𝐽 ∈ 𝑛-Locally Comp β†’ 𝐽 ∈ Top)
2 resttop 22664 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ (𝐽 β†Ύt 𝐴) ∈ Top)
31, 2sylan 581 . 2 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ (𝐽 β†Ύt 𝐴) ∈ Top)
4 elrest 17373 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) β†’ (π‘₯ ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ 𝐴)))
5 simpll 766 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ 𝐽 ∈ 𝑛-Locally Comp)
6 simprl 770 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ 𝑒 ∈ 𝐽)
7 simprr 772 . . . . . . . . . . 11 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ 𝑦 ∈ (𝑒 ∩ 𝐴))
87elin1d 4199 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsdβ€˜π½)) ∧ (𝑒 ∈ 𝐽 ∧ 𝑦 ∈ (𝑒 ∩ 𝐴))) β†’ 𝑦 ∈ 𝑒)
9 nlly2i 22980 . . . . . . . . . 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 1372 . . . . . . . . 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 725 . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . 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 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†’ 𝐴 ∈ (Clsdβ€˜π½))
14 simprlr 779 . . . . . . . . . . . . . . . 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 17374 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsdβ€˜π½) ∧ 𝑀 ∈ 𝐽) β†’ (𝑀 ∩ 𝐴) ∈ (𝐽 β†Ύt 𝐴))
1612, 13, 14, 15syl3anc 1372 . . . . . . . . . . . . . . 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 1222 . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . 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 4200 . . . . . . . . . . . . . . . 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 4195 . . . . . . . . . . . . . . 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 22623 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 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 1223 . . . . . . . . . . . . . . 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 4236 . . . . . . . . . . . . . 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 4230 . . . . . . . . . . . . . . 15 (𝑠 ∩ 𝐴) βŠ† 𝐴
26 eqid 2733 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝐽 = βˆͺ 𝐽
2726cldss 22533 . . . . . . . . . . . . . . . . 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 22666 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴 βŠ† βˆͺ 𝐽) β†’ 𝐴 = βˆͺ (𝐽 β†Ύt 𝐴))
3012, 28, 29syl2anc 585 . . . . . . . . . . . . . . 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 4035 . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . 15 βˆͺ (𝐽 β†Ύt 𝐴) = βˆͺ (𝐽 β†Ύt 𝐴)
3332ssnei2 22620 . . . . . . . . . . . . . 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 838 . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . 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 4612 . . . . . . . . . . . . . . 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 4236 . . . . . . . . . . . . . 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 3479 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
3938inex1 5318 . . . . . . . . . . . . . . 15 (𝑠 ∩ 𝐴) ∈ V
4039elpw 4607 . . . . . . . . . . . . . 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 4195 . . . . . . . . . . . 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 22669 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 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 4229 . . . . . . . . . . . . . . . 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 22669 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . 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 1224 . . . . . . . . . . . . . 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 4202 . . . . . . . . . . . . . . 15 (𝑠 ∩ 𝐴) = (𝐴 ∩ 𝑠)
53 eqid 2733 . . . . . . . . . . . . . . . . 17 (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)
54 ineq1 4206 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 β†’ (𝑣 ∩ 𝑠) = (𝐴 ∩ 𝑠))
5554rspceeqv 3634 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (Clsdβ€˜π½) ∧ (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)) β†’ βˆƒπ‘£ ∈ (Clsdβ€˜π½)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠))
5613, 53, 55sylancl 587 . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . 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 4942 . . . . . . . . . . . . . . . . . . 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 3993 . . . . . . . . . . . . . . . . 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 22676 . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . 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 2838 . . . . . . . . . . . . . 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 22906 . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . 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 2834 . . . . . . . . . . . 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 7417 . . . . . . . . . . . . . 14 (𝑣 = (𝑠 ∩ 𝐴) β†’ ((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) = ((𝐽 β†Ύt 𝐴) β†Ύt (𝑠 ∩ 𝐴)))
6968eleq1d 2819 . . . . . . . . . . . . 13 (𝑣 = (𝑠 ∩ 𝐴) β†’ (((𝐽 β†Ύt 𝐴) β†Ύt 𝑣) ∈ Comp ↔ ((𝐽 β†Ύt 𝐴) β†Ύt (𝑠 ∩ 𝐴)) ∈ Comp))
7069rspcev 3613 . . . . . . . . . . . 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 585 . . . . . . . . . . 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 458 . . . . . . . . . 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 3212 . . . . . . . . 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 469 . . . . . . 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 3147 . . . . . 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 4617 . . . . . . . . 9 (π‘₯ = (𝑒 ∩ 𝐴) β†’ 𝒫 π‘₯ = 𝒫 (𝑒 ∩ 𝐴))
7877ineq2d 4213 . . . . . . . 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 3327 . . . . . . 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 3334 . . . . . 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 3156 . . . 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 3146 . 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 22973 . 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 584 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409   β†Ύt crest 17366  Topctop 22395  Clsdccld 22520  neicnei 22601  Compccmp 22890  π‘›-Locally cnlly 22969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9406  df-rest 17368  df-topgen 17389  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-nei 22602  df-cmp 22891  df-nlly 22971
This theorem is referenced by:  rellycmp  24473
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