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Theorem llycmpkgen2 21547
Description: A locally compact space is compactly generated. (This variant of llycmpkgen 21549 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 4611 . . . . . . . . . . 11 (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢 (𝑘Gen‘𝐽))
32adantl 473 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢 (𝑘Gen‘𝐽))
4 iskgen3.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
54kgenuni 21536 . . . . . . . . . . . 12 (𝐽 ∈ Top → 𝑋 = (𝑘Gen‘𝐽))
61, 5syl 17 . . . . . . . . . . 11 (𝜑𝑋 = (𝑘Gen‘𝐽))
76adantr 472 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑋 = (𝑘Gen‘𝐽))
83, 7sseqtr4d 3775 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢𝑋)
98sselda 3736 . . . . . . . 8 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → 𝑥𝑋)
10 llycmpkgen2.3 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
1110adantlr 753 . . . . . . . 8 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
129, 11syldan 488 . . . . . . 7 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)
131ad3antrrr 768 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝐽 ∈ Top)
14 difss 3872 . . . . . . . . . 10 (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋
154ntropn 21047 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽)
1613, 14, 15sylancl 697 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽)
17 simprl 811 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘 ∈ ((nei‘𝐽)‘{𝑥}))
184neii1 21104 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑘 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑘𝑋)
1913, 17, 18syl2anc 696 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘𝑋)
204ntropn 21047 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑘𝑋) → ((int‘𝐽)‘𝑘) ∈ 𝐽)
2113, 19, 20syl2anc 696 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ∈ 𝐽)
22 inopn 20898 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∈ 𝐽 ∧ ((int‘𝐽)‘𝑘) ∈ 𝐽) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽)
2313, 16, 21, 22syl3anc 1473 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽)
24 inss1 3968 . . . . . . . . . . 11 (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘) ⊆ ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘)))
25 simplr 809 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑢)
264ntrss2 21055 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑘𝑋) → ((int‘𝐽)‘𝑘) ⊆ 𝑘)
2713, 19, 26syl2anc 696 . . . . . . . . . . . . . 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 4477 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → {𝑥} ⊆ 𝑋)
304neiint 21102 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝑋𝑘𝑋) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘)))
3113, 29, 19, 30syl3anc 1473 . . . . . . . . . . . . . . . 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 3335 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3433snss 4452 . . . . . . . . . . . . . . 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 3737 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥𝑘)
3725, 36elind 3933 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑢𝑘))
38 simpllr 817 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑢 ∈ (𝑘Gen‘𝐽))
39 simprr 813 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Comp)
40 kgeni 21534 . . . . . . . . . . . . . . 15 ((𝑢 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑢𝑘) ∈ (𝐽t 𝑘))
4138, 39, 40syl2anc 696 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ∈ (𝐽t 𝑘))
42 vex 3335 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
43 resttop 21158 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝑘 ∈ V) → (𝐽t 𝑘) ∈ Top)
4413, 42, 43sylancl 697 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
45 inss2 3969 . . . . . . . . . . . . . . . 16 (𝑢𝑘) ⊆ 𝑘
464restuni 21160 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑘𝑋) → 𝑘 = (𝐽t 𝑘))
4713, 19, 46syl2anc 696 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘 = (𝐽t 𝑘))
4845, 47syl5sseq 3786 . . . . . . . . . . . . . . 15 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ⊆ (𝐽t 𝑘))
49 eqid 2752 . . . . . . . . . . . . . . . 16 (𝐽t 𝑘) = (𝐽t 𝑘)
5049isopn3 21064 . . . . . . . . . . . . . . 15 (((𝐽t 𝑘) ∈ Top ∧ (𝑢𝑘) ⊆ (𝐽t 𝑘)) → ((𝑢𝑘) ∈ (𝐽t 𝑘) ↔ ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (𝑢𝑘)))
5144, 48, 50syl2anc 696 . . . . . . . . . . . . . 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 2752 . . . . . . . . . . . . . . 15 (𝐽t 𝑘) = (𝐽t 𝑘)
554, 54restntr 21180 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑘𝑋 ∧ (𝑢𝑘) ⊆ 𝑘) → ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5613, 19, 53, 55syl3anc 1473 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘(𝐽t 𝑘))‘(𝑢𝑘)) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5752, 56eqtr3d 2788 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) = (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5837, 57eleqtrd 2833 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) ∩ 𝑘))
5924, 58sseldi 3734 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))))
60 undif3 4023 . . . . . . . . . . . . 13 ((𝑢𝑘) ∪ (𝑋𝑘)) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢𝑘)))
61 incom 3940 . . . . . . . . . . . . . . . 16 (𝑢𝑘) = (𝑘𝑢)
6261difeq2i 3860 . . . . . . . . . . . . . . 15 (𝑘 ∖ (𝑢𝑘)) = (𝑘 ∖ (𝑘𝑢))
63 difin 3996 . . . . . . . . . . . . . . 15 (𝑘 ∖ (𝑘𝑢)) = (𝑘𝑢)
6462, 63eqtri 2774 . . . . . . . . . . . . . 14 (𝑘 ∖ (𝑢𝑘)) = (𝑘𝑢)
6564difeq2i 3860 . . . . . . . . . . . . 13 (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢𝑘))) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢))
6660, 65eqtri 2774 . . . . . . . . . . . 12 ((𝑢𝑘) ∪ (𝑋𝑘)) = (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢))
6745, 19syl5ss 3747 . . . . . . . . . . . . . 14 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑢𝑘) ⊆ 𝑋)
68 ssequn1 3918 . . . . . . . . . . . . . 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 3862 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((𝑢𝑘) ∪ 𝑋) ∖ (𝑘𝑢)) = (𝑋 ∖ (𝑘𝑢)))
7166, 70syl5eq 2798 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑢𝑘) ∪ (𝑋𝑘)) = (𝑋 ∖ (𝑘𝑢)))
7271fveq2d 6348 . . . . . . . . . 10 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘((𝑢𝑘) ∪ (𝑋𝑘))) = ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))))
7359, 72eleqtrd 2833 . . . . . . . . 9 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))))
7473, 35elind 3933 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)))
75 sslin 3974 . . . . . . . . . 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 21055 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢)))
7813, 14, 77sylancl 697 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ (𝑋 ∖ (𝑘𝑢)))
7978difss2d 3875 . . . . . . . . . . . 12 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ⊆ 𝑋)
80 reldisj 4155 . . . . . . . . . . . 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 4082 . . . . . . . . . 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 3746 . . . . . . . 8 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)
86 eleq2 2820 . . . . . . . . . 10 (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑥𝑧𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘))))
87 sseq1 3759 . . . . . . . . . 10 (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑧𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢))
8886, 87anbi12d 749 . . . . . . . . 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 3441 . . . . . . . 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 1471 . . . . . . 7 ((((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽t 𝑘) ∈ Comp)) → ∃𝑧𝐽 (𝑥𝑧𝑧𝑢))
9112, 90rexlimddv 3165 . . . . . 6 (((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥𝑢) → ∃𝑧𝐽 (𝑥𝑧𝑧𝑢))
9291ralrimiva 3096 . . . . 5 ((𝜑𝑢 ∈ (𝑘Gen‘𝐽)) → ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢))
9392ex 449 . . . 4 (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
94 eltop2 20973 . . . . 5 (𝐽 ∈ Top → (𝑢𝐽 ↔ ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
951, 94syl 17 . . . 4 (𝜑 → (𝑢𝐽 ↔ ∀𝑥𝑢𝑧𝐽 (𝑥𝑧𝑧𝑢)))
9693, 95sylibrd 249 . . 3 (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢𝐽))
9796ssrdv 3742 . 2 (𝜑 → (𝑘Gen‘𝐽) ⊆ 𝐽)
98 iskgen2 21545 . 2 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
991, 97, 98sylanbrc 701 1 (𝜑𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042  wrex 3043  Vcvv 3332  cdif 3704  cun 3705  cin 3706  wss 3707  c0 4050  {csn 4313   cuni 4580  ran crn 5259  cfv 6041  (class class class)co 6805  t crest 16275  Topctop 20892  intcnt 21015  neicnei 21095  Compccmp 21383  𝑘Genckgen 21530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-oadd 7725  df-er 7903  df-en 8114  df-fin 8117  df-fi 8474  df-rest 16277  df-topgen 16298  df-top 20893  df-topon 20910  df-bases 20944  df-ntr 21018  df-nei 21096  df-cmp 21384  df-kgen 21531
This theorem is referenced by:  cmpkgen  21548  llycmpkgen  21549
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