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Theorem llycmpkgen2 21633
Description: A locally compact space is compactly generated. (This variant of llycmpkgen 21635 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 4625 . . . . . . . . . . 11 (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢 (𝑘Gen‘𝐽))
32adantl 473 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢 (𝑘Gen‘𝐽))
4 iskgen3.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
54kgenuni 21622 . . . . . . . . . . . 12 (𝐽 ∈ Top → 𝑋 = (𝑘Gen‘𝐽))
61, 5syl 17 . . . . . . . . . . 11 (𝜑𝑋 = (𝑘Gen‘𝐽))
76adantr 472 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑋 = (𝑘Gen‘𝐽))
83, 7sseqtr4d 3802 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢𝑋)
98sselda 3761 . . . . . . . 8 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → 𝑥𝑋)
10 llycmpkgen2.3 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
1110adantlr 706 . . . . . . . 8 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
129, 11syldan 585 . . . . . . 7 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
131ad3antrrr 721 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝐽 ∈ Top)
14 difss 3899 . . . . . . . . . 10 (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋
154ntropn 21133 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽)
1613, 14, 15sylancl 580 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽)
17 simprl 787 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘 ∈ ((nei‘𝐽)‘{𝑥}))
184neii1 21190 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑘 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑘𝑋)
1913, 17, 18syl2anc 579 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘𝑋)
204ntropn 21133 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑘𝑋) → ((int‘𝐽)‘𝑘) ∈ 𝐽)
2113, 19, 20syl2anc 579 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ∈ 𝐽)
22 inopn 20983 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽 ∧ ((int‘𝐽)‘𝑘) ∈ 𝐽) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽)
2313, 16, 21, 22syl3anc 1490 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽)
24 inss1 3992 . . . . . . . . . . 11 (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘) ⊆ ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘)))
25 simplr 785 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑢)
264ntrss2 21141 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑘𝑋) → ((int‘𝐽)‘𝑘) ⊆ 𝑘)
2713, 19, 26syl2anc 579 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ⊆ 𝑘)
289adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑋)
2928snssd 4494 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → {𝑥} ⊆ 𝑋)
304neiint 21188 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝑋𝑘𝑋) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘)))
3113, 29, 19, 30syl3anc 1490 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘)))
3217, 31mpbid 223 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → {𝑥} ⊆ ((int‘𝐽)‘𝑘))
33 vex 3353 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3433snss 4470 . . . . . . . . . . . . . . 15 (𝑥 ∈ ((int‘𝐽)‘𝑘) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘))
3532, 34sylibr 225 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘𝑘))
3627, 35sseldd 3762 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑘)
3725, 36elind 3960 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑢𝑘))
38 simpllr 793 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑢 ∈ (𝑘Gen‘𝐽))
39 simprr 789 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Comp)
40 kgeni 21620 . . . . . . . . . . . . . . 15 ((𝑢 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑢𝑘) ∈ (𝐽t 𝑘))
4138, 39, 40syl2anc 579 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ∈ (𝐽t 𝑘))
42 vex 3353 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
43 resttop 21244 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝑘 ∈ V) → (𝐽t 𝑘) ∈ Top)
4413, 42, 43sylancl 580 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
45 inss2 3993 . . . . . . . . . . . . . . . 16 (𝑢𝑘) ⊆ 𝑘
464restuni 21246 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑘𝑋) → 𝑘 = (𝐽t 𝑘))
4713, 19, 46syl2anc 579 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘 = (𝐽t 𝑘))
4845, 47syl5sseq 3813 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ⊆ (𝐽t 𝑘))
49 eqid 2765 . . . . . . . . . . . . . . . 16 (𝐽t 𝑘) = (𝐽t 𝑘)
5049isopn3 21150 . . . . . . . . . . . . . . 15 (((𝐽t 𝑘) ∈ Top ∧ (𝑢𝑘) ⊆ (𝐽t 𝑘)) → ((𝑢𝑘) ∈ (𝐽t 𝑘) ↔ ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (𝑢𝑘)))
5144, 48, 50syl2anc 579 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑢𝑘) ∈ (𝐽t 𝑘) ↔ ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (𝑢𝑘)))
5241, 51mpbid 223 . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . . 15 (𝐽t 𝑘) = (𝐽t 𝑘)
554, 54restntr 21266 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑘𝑋 ∧ (𝑢𝑘) ⊆ 𝑘) → ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5613, 19, 53, 55syl3anc 1490 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5752, 56eqtr3d 2801 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5837, 57eleqtrd 2846 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5924, 58sseldi 3759 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))))
60 undif3 4053 . . . . . . . . . . . . 13 ((𝑢𝑘) ∪ (𝑋𝑘)) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢𝑘)))
61 incom 3967 . . . . . . . . . . . . . . . 16 (𝑢𝑘) = (𝑘𝑢)
6261difeq2i 3887 . . . . . . . . . . . . . . 15 (𝑘 ∖ (𝑢𝑘)) = (𝑘 ∖ (𝑘𝑢))
63 difin 4026 . . . . . . . . . . . . . . 15 (𝑘 ∖ (𝑘𝑢)) = (𝑘𝑢)
6462, 63eqtri 2787 . . . . . . . . . . . . . 14 (𝑘 ∖ (𝑢𝑘)) = (𝑘𝑢)
6564difeq2i 3887 . . . . . . . . . . . . 13 (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢𝑘))) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢))
6660, 65eqtri 2787 . . . . . . . . . . . 12 ((𝑢𝑘) ∪ (𝑋𝑘)) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢))
6745, 19syl5ss 3772 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ⊆ 𝑋)
68 ssequn1 3945 . . . . . . . . . . . . . 14 ((𝑢𝑘) ⊆ 𝑋 ↔ ((𝑢𝑘) ∪ 𝑋) = 𝑋)
6967, 68sylib 209 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑢𝑘) ∪ 𝑋) = 𝑋)
7069difeq1d 3889 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢)) = (𝑋 ∖ (𝑘𝑢)))
7166, 70syl5eq 2811 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑢𝑘) ∪ (𝑋𝑘)) = (𝑋 ∖ (𝑘𝑢)))
7271fveq2d 6379 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) = ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))))
7359, 72eleqtrd 2846 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))))
7473, 35elind 3960 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)))
75 sslin 3998 . . . . . . . . . 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 21141 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢)))
7813, 14, 77sylancl 580 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢)))
7978difss2d 3902 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ 𝑋)
80 reldisj 4181 . . . . . . . . . . . 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 248 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ (𝑘𝑢)) = ∅)
83 inssdif0 4112 . . . . . . . . . 10 ((((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ 𝑘) ⊆ 𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ (𝑘𝑢)) = ∅)
8482, 83sylibr 225 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ 𝑘) ⊆ 𝑢)
8576, 84sstrd 3771 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)
86 eleq2 2833 . . . . . . . . . 10 (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑥𝑧𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘))))
87 sseq1 3786 . . . . . . . . . 10 (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑧𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢))
8886, 87anbi12d 624 . . . . . . . . 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 3461 . . . . . . . 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 865 . . . . . . 7 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ∃𝑧𝐽 (𝑥𝑧𝑧𝑢))
9112, 90rexlimddv 3182 . . . . . 6 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → ∃𝑧𝐽 (𝑥𝑧𝑧𝑢))
9291ralrimiva 3113 . . . . 5 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢))
9392ex 401 . . . 4 (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
94 eltop2 21059 . . . . 5 (𝐽 ∈ Top → (𝑢𝐽 ↔ ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
951, 94syl 17 . . . 4 (𝜑 → (𝑢𝐽 ↔ ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
9693, 95sylibrd 250 . . 3 (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢𝐽))
9796ssrdv 3767 . 2 (𝜑 → (𝑘Gen‘𝐽) ⊆ 𝐽)
98 iskgen2 21631 . 2 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
991, 97, 98sylanbrc 578 1 (𝜑𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334   cuni 4594  ran crn 5278  cfv 6068  (class class class)co 6842  t crest 16349  Topctop 20977  intcnt 21101  neicnei 21181  Compccmp 21469  𝑘Genckgen 21616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768  df-er 7947  df-en 8161  df-fin 8164  df-fi 8524  df-rest 16351  df-topgen 16372  df-top 20978  df-topon 20995  df-bases 21030  df-ntr 21104  df-nei 21182  df-cmp 21470  df-kgen 21617
This theorem is referenced by:  cmpkgen  21634  llycmpkgen  21635
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