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Theorem llycmpkgen2 21574
Description: A locally compact space is compactly generated. (This variant of llycmpkgen 21576 uses the weaker definition of locally compact, "every point has a compact neighborhood", instead of "every point has a local base of compact neighborhoods".) (Contributed by Mario Carneiro, 21-Mar-2015.)
Hypotheses
Ref Expression
iskgen3.1 𝑋 = 𝐽
llycmpkgen2.2 (𝜑𝐽 ∈ Top)
llycmpkgen2.3 ((𝜑𝑥𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
Assertion
Ref Expression
llycmpkgen2 (𝜑𝐽 ∈ ran 𝑘Gen)
Distinct variable groups:   𝑥,𝑘,𝐽   𝜑,𝑘,𝑥   𝑘,𝑋
Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem llycmpkgen2
Dummy variables 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llycmpkgen2.2 . 2 (𝜑𝐽 ∈ Top)
2 elssuni 4603 . . . . . . . . . . 11 (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢 (𝑘Gen‘𝐽))
32adantl 467 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢 (𝑘Gen‘𝐽))
4 iskgen3.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
54kgenuni 21563 . . . . . . . . . . . 12 (𝐽 ∈ Top → 𝑋 = (𝑘Gen‘𝐽))
61, 5syl 17 . . . . . . . . . . 11 (𝜑𝑋 = (𝑘Gen‘𝐽))
76adantr 466 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑋 = (𝑘Gen‘𝐽))
83, 7sseqtr4d 3791 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢𝑋)
98sselda 3752 . . . . . . . 8 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → 𝑥𝑋)
10 llycmpkgen2.3 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
1110adantlr 694 . . . . . . . 8 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
129, 11syldan 579 . . . . . . 7 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
131ad3antrrr 709 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝐽 ∈ Top)
14 difss 3888 . . . . . . . . . 10 (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋
154ntropn 21074 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽)
1613, 14, 15sylancl 574 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽)
17 simprl 754 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘 ∈ ((nei‘𝐽)‘{𝑥}))
184neii1 21131 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑘 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑘𝑋)
1913, 17, 18syl2anc 573 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘𝑋)
204ntropn 21074 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑘𝑋) → ((int‘𝐽)‘𝑘) ∈ 𝐽)
2113, 19, 20syl2anc 573 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ∈ 𝐽)
22 inopn 20924 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽 ∧ ((int‘𝐽)‘𝑘) ∈ 𝐽) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽)
2313, 16, 21, 22syl3anc 1476 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽)
24 inss1 3981 . . . . . . . . . . 11 (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘) ⊆ ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘)))
25 simplr 752 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑢)
264ntrss2 21082 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑘𝑋) → ((int‘𝐽)‘𝑘) ⊆ 𝑘)
2713, 19, 26syl2anc 573 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ⊆ 𝑘)
289adantr 466 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑋)
2928snssd 4475 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → {𝑥} ⊆ 𝑋)
304neiint 21129 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝑋𝑘𝑋) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘)))
3113, 29, 19, 30syl3anc 1476 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘)))
3217, 31mpbid 222 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → {𝑥} ⊆ ((int‘𝐽)‘𝑘))
33 vex 3354 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3433snss 4451 . . . . . . . . . . . . . . 15 (𝑥 ∈ ((int‘𝐽)‘𝑘) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘))
3532, 34sylibr 224 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘𝑘))
3627, 35sseldd 3753 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑘)
3725, 36elind 3949 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑢𝑘))
38 simpllr 760 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑢 ∈ (𝑘Gen‘𝐽))
39 simprr 756 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Comp)
40 kgeni 21561 . . . . . . . . . . . . . . 15 ((𝑢 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑢𝑘) ∈ (𝐽t 𝑘))
4138, 39, 40syl2anc 573 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ∈ (𝐽t 𝑘))
42 vex 3354 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
43 resttop 21185 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝑘 ∈ V) → (𝐽t 𝑘) ∈ Top)
4413, 42, 43sylancl 574 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
45 inss2 3982 . . . . . . . . . . . . . . . 16 (𝑢𝑘) ⊆ 𝑘
464restuni 21187 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑘𝑋) → 𝑘 = (𝐽t 𝑘))
4713, 19, 46syl2anc 573 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘 = (𝐽t 𝑘))
4845, 47syl5sseq 3802 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ⊆ (𝐽t 𝑘))
49 eqid 2771 . . . . . . . . . . . . . . . 16 (𝐽t 𝑘) = (𝐽t 𝑘)
5049isopn3 21091 . . . . . . . . . . . . . . 15 (((𝐽t 𝑘) ∈ Top ∧ (𝑢𝑘) ⊆ (𝐽t 𝑘)) → ((𝑢𝑘) ∈ (𝐽t 𝑘) ↔ ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (𝑢𝑘)))
5144, 48, 50syl2anc 573 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑢𝑘) ∈ (𝐽t 𝑘) ↔ ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (𝑢𝑘)))
5241, 51mpbid 222 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (𝑢𝑘))
5345a1i 11 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ⊆ 𝑘)
54 eqid 2771 . . . . . . . . . . . . . . 15 (𝐽t 𝑘) = (𝐽t 𝑘)
554, 54restntr 21207 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑘𝑋 ∧ (𝑢𝑘) ⊆ 𝑘) → ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5613, 19, 53, 55syl3anc 1476 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5752, 56eqtr3d 2807 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5837, 57eleqtrd 2852 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5924, 58sseldi 3750 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))))
60 undif3 4037 . . . . . . . . . . . . 13 ((𝑢𝑘) ∪ (𝑋𝑘)) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢𝑘)))
61 incom 3956 . . . . . . . . . . . . . . . 16 (𝑢𝑘) = (𝑘𝑢)
6261difeq2i 3876 . . . . . . . . . . . . . . 15 (𝑘 ∖ (𝑢𝑘)) = (𝑘 ∖ (𝑘𝑢))
63 difin 4010 . . . . . . . . . . . . . . 15 (𝑘 ∖ (𝑘𝑢)) = (𝑘𝑢)
6462, 63eqtri 2793 . . . . . . . . . . . . . 14 (𝑘 ∖ (𝑢𝑘)) = (𝑘𝑢)
6564difeq2i 3876 . . . . . . . . . . . . 13 (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢𝑘))) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢))
6660, 65eqtri 2793 . . . . . . . . . . . 12 ((𝑢𝑘) ∪ (𝑋𝑘)) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢))
6745, 19syl5ss 3763 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ⊆ 𝑋)
68 ssequn1 3934 . . . . . . . . . . . . . 14 ((𝑢𝑘) ⊆ 𝑋 ↔ ((𝑢𝑘) ∪ 𝑋) = 𝑋)
6967, 68sylib 208 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑢𝑘) ∪ 𝑋) = 𝑋)
7069difeq1d 3878 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢)) = (𝑋 ∖ (𝑘𝑢)))
7166, 70syl5eq 2817 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑢𝑘) ∪ (𝑋𝑘)) = (𝑋 ∖ (𝑘𝑢)))
7271fveq2d 6336 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) = ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))))
7359, 72eleqtrd 2852 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))))
7473, 35elind 3949 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)))
75 sslin 3987 . . . . . . . . . 10 (((int‘𝐽)‘𝑘) ⊆ 𝑘 → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ 𝑘))
7627, 75syl 17 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ 𝑘))
774ntrss2 21082 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢)))
7813, 14, 77sylancl 574 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢)))
7978difss2d 3891 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ 𝑋)
80 reldisj 4163 . . . . . . . . . . . 12 (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ 𝑋 → ((((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ (𝑘𝑢)) = ∅ ↔ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢))))
8179, 80syl 17 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ (𝑘𝑢)) = ∅ ↔ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢))))
8278, 81mpbird 247 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ (𝑘𝑢)) = ∅)
83 inssdif0 4094 . . . . . . . . . 10 ((((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ 𝑘) ⊆ 𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ (𝑘𝑢)) = ∅)
8482, 83sylibr 224 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ 𝑘) ⊆ 𝑢)
8576, 84sstrd 3762 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)
86 eleq2 2839 . . . . . . . . . 10 (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑥𝑧𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘))))
87 sseq1 3775 . . . . . . . . . 10 (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑧𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢))
8886, 87anbi12d 616 . . . . . . . . 9 (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) → ((𝑥𝑧𝑧𝑢) ↔ (𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∧ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)))
8988rspcev 3460 . . . . . . . 8 (((((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽 ∧ (𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∧ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)) → ∃𝑧𝐽 (𝑥𝑧𝑧𝑢))
9023, 74, 85, 89syl12anc 1474 . . . . . . 7 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ∃𝑧𝐽 (𝑥𝑧𝑧𝑢))
9112, 90rexlimddv 3183 . . . . . 6 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → ∃𝑧𝐽 (𝑥𝑧𝑧𝑢))
9291ralrimiva 3115 . . . . 5 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢))
9392ex 397 . . . 4 (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
94 eltop2 21000 . . . . 5 (𝐽 ∈ Top → (𝑢𝐽 ↔ ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
951, 94syl 17 . . . 4 (𝜑 → (𝑢𝐽 ↔ ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
9693, 95sylibrd 249 . . 3 (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢𝐽))
9796ssrdv 3758 . 2 (𝜑 → (𝑘Gen‘𝐽) ⊆ 𝐽)
98 iskgen2 21572 . 2 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
991, 97, 98sylanbrc 572 1 (𝜑𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  {csn 4316   cuni 4574  ran crn 5250  cfv 6031  (class class class)co 6793  t crest 16289  Topctop 20918  intcnt 21042  neicnei 21122  Compccmp 21410  𝑘Genckgen 21557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-er 7896  df-en 8110  df-fin 8113  df-fi 8473  df-rest 16291  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-ntr 21045  df-nei 21123  df-cmp 21411  df-kgen 21558
This theorem is referenced by:  cmpkgen  21575  llycmpkgen  21576
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