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Theorem llycmpkgen2 21842
Description: A locally compact space is compactly generated. (This variant of llycmpkgen 21844 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 4774 . . . . . . . . . . 11 (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢 (𝑘Gen‘𝐽))
32adantl 482 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢 (𝑘Gen‘𝐽))
4 iskgen3.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
54kgenuni 21831 . . . . . . . . . . . 12 (𝐽 ∈ Top → 𝑋 = (𝑘Gen‘𝐽))
61, 5syl 17 . . . . . . . . . . 11 (𝜑𝑋 = (𝑘Gen‘𝐽))
76adantr 481 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑋 = (𝑘Gen‘𝐽))
83, 7sseqtr4d 3929 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢𝑋)
98sselda 3889 . . . . . . . 8 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → 𝑥𝑋)
10 llycmpkgen2.3 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
1110adantlr 711 . . . . . . . 8 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
129, 11syldan 591 . . . . . . 7 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
131ad3antrrr 726 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝐽 ∈ Top)
14 difss 4029 . . . . . . . . . 10 (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋
154ntropn 21341 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽)
1613, 14, 15sylancl 586 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽)
17 simprl 767 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘 ∈ ((nei‘𝐽)‘{𝑥}))
184neii1 21398 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑘 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑘𝑋)
1913, 17, 18syl2anc 584 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘𝑋)
204ntropn 21341 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑘𝑋) → ((int‘𝐽)‘𝑘) ∈ 𝐽)
2113, 19, 20syl2anc 584 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ∈ 𝐽)
22 inopn 21191 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽 ∧ ((int‘𝐽)‘𝑘) ∈ 𝐽) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽)
2313, 16, 21, 22syl3anc 1364 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽)
24 simplr 765 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑢)
254ntrss2 21349 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑘𝑋) → ((int‘𝐽)‘𝑘) ⊆ 𝑘)
2613, 19, 25syl2anc 584 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ⊆ 𝑘)
279adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑋)
2827snssd 4649 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → {𝑥} ⊆ 𝑋)
294neiint 21396 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝑋𝑘𝑋) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘)))
3013, 28, 19, 29syl3anc 1364 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘)))
3117, 30mpbid 233 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → {𝑥} ⊆ ((int‘𝐽)‘𝑘))
32 vex 3440 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3332snss 4625 . . . . . . . . . . . . . . 15 (𝑥 ∈ ((int‘𝐽)‘𝑘) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘))
3431, 33sylibr 235 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘𝑘))
3526, 34sseldd 3890 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑘)
3624, 35elind 4092 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑢𝑘))
37 simpllr 772 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑢 ∈ (𝑘Gen‘𝐽))
38 simprr 769 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Comp)
39 kgeni 21829 . . . . . . . . . . . . . . 15 ((𝑢 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑢𝑘) ∈ (𝐽t 𝑘))
4037, 38, 39syl2anc 584 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ∈ (𝐽t 𝑘))
41 vex 3440 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
42 resttop 21452 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝑘 ∈ V) → (𝐽t 𝑘) ∈ Top)
4313, 41, 42sylancl 586 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
44 inss2 4126 . . . . . . . . . . . . . . . 16 (𝑢𝑘) ⊆ 𝑘
454restuni 21454 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑘𝑋) → 𝑘 = (𝐽t 𝑘))
4613, 19, 45syl2anc 584 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘 = (𝐽t 𝑘))
4744, 46sseqtrid 3940 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ⊆ (𝐽t 𝑘))
48 eqid 2795 . . . . . . . . . . . . . . . 16 (𝐽t 𝑘) = (𝐽t 𝑘)
4948isopn3 21358 . . . . . . . . . . . . . . 15 (((𝐽t 𝑘) ∈ Top ∧ (𝑢𝑘) ⊆ (𝐽t 𝑘)) → ((𝑢𝑘) ∈ (𝐽t 𝑘) ↔ ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (𝑢𝑘)))
5043, 47, 49syl2anc 584 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑢𝑘) ∈ (𝐽t 𝑘) ↔ ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (𝑢𝑘)))
5140, 50mpbid 233 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (𝑢𝑘))
5244a1i 11 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ⊆ 𝑘)
53 eqid 2795 . . . . . . . . . . . . . . 15 (𝐽t 𝑘) = (𝐽t 𝑘)
544, 53restntr 21474 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑘𝑋 ∧ (𝑢𝑘) ⊆ 𝑘) → ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5513, 19, 52, 54syl3anc 1364 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5651, 55eqtr3d 2833 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5736, 56eleqtrd 2885 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5857elin1d 4096 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))))
59 undif3 4185 . . . . . . . . . . . . 13 ((𝑢𝑘) ∪ (𝑋𝑘)) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢𝑘)))
60 incom 4099 . . . . . . . . . . . . . . . 16 (𝑢𝑘) = (𝑘𝑢)
6160difeq2i 4017 . . . . . . . . . . . . . . 15 (𝑘 ∖ (𝑢𝑘)) = (𝑘 ∖ (𝑘𝑢))
62 difin 4158 . . . . . . . . . . . . . . 15 (𝑘 ∖ (𝑘𝑢)) = (𝑘𝑢)
6361, 62eqtri 2819 . . . . . . . . . . . . . 14 (𝑘 ∖ (𝑢𝑘)) = (𝑘𝑢)
6463difeq2i 4017 . . . . . . . . . . . . 13 (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢𝑘))) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢))
6559, 64eqtri 2819 . . . . . . . . . . . 12 ((𝑢𝑘) ∪ (𝑋𝑘)) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢))
6644, 19syl5ss 3900 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ⊆ 𝑋)
67 ssequn1 4077 . . . . . . . . . . . . . 14 ((𝑢𝑘) ⊆ 𝑋 ↔ ((𝑢𝑘) ∪ 𝑋) = 𝑋)
6866, 67sylib 219 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑢𝑘) ∪ 𝑋) = 𝑋)
6968difeq1d 4019 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢)) = (𝑋 ∖ (𝑘𝑢)))
7065, 69syl5eq 2843 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑢𝑘) ∪ (𝑋𝑘)) = (𝑋 ∖ (𝑘𝑢)))
7170fveq2d 6542 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) = ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))))
7258, 71eleqtrd 2885 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))))
7372, 34elind 4092 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)))
74 sslin 4131 . . . . . . . . . 10 (((int‘𝐽)‘𝑘) ⊆ 𝑘 → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ 𝑘))
7526, 74syl 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))
764ntrss2 21349 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢)))
7713, 14, 76sylancl 586 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢)))
7877difss2d 4032 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ 𝑋)
79 reldisj 4316 . . . . . . . . . . . 12 (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ 𝑋 → ((((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ (𝑘𝑢)) = ∅ ↔ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢))))
8078, 79syl 17 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ (𝑘𝑢)) = ∅ ↔ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢))))
8177, 80mpbird 258 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ (𝑘𝑢)) = ∅)
82 inssdif0 4249 . . . . . . . . . 10 ((((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ 𝑘) ⊆ 𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ (𝑘𝑢)) = ∅)
8381, 82sylibr 235 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ 𝑘) ⊆ 𝑢)
8475, 83sstrd 3899 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)
85 eleq2 2871 . . . . . . . . . 10 (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑥𝑧𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘))))
86 sseq1 3913 . . . . . . . . . 10 (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑧𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢))
8785, 86anbi12d 630 . . . . . . . . 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)))
8887rspcev 3559 . . . . . . . 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))
8923, 73, 84, 88syl12anc 833 . . . . . . 7 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ∃𝑧𝐽 (𝑥𝑧𝑧𝑢))
9012, 89rexlimddv 3254 . . . . . 6 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → ∃𝑧𝐽 (𝑥𝑧𝑧𝑢))
9190ralrimiva 3149 . . . . 5 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢))
9291ex 413 . . . 4 (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
93 eltop2 21267 . . . . 5 (𝐽 ∈ Top → (𝑢𝐽 ↔ ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
941, 93syl 17 . . . 4 (𝜑 → (𝑢𝐽 ↔ ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
9592, 94sylibrd 260 . . 3 (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢𝐽))
9695ssrdv 3895 . 2 (𝜑 → (𝑘Gen‘𝐽) ⊆ 𝐽)
97 iskgen2 21840 . 2 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
981, 96, 97sylanbrc 583 1 (𝜑𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  wrex 3106  Vcvv 3437  cdif 3856  cun 3857  cin 3858  wss 3859  c0 4211  {csn 4472   cuni 4745  ran crn 5444  cfv 6225  (class class class)co 7016  t crest 16523  Topctop 21185  intcnt 21309  neicnei 21389  Compccmp 21678  𝑘Genckgen 21825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-oadd 7957  df-er 8139  df-en 8358  df-fin 8361  df-fi 8721  df-rest 16525  df-topgen 16546  df-top 21186  df-topon 21203  df-bases 21238  df-ntr 21312  df-nei 21390  df-cmp 21679  df-kgen 21826
This theorem is referenced by:  cmpkgen  21843  llycmpkgen  21844
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