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Theorem kgencn3 23053
Description: The set of continuous functions from 𝐽 to 𝐾 is unaffected by k-ification of 𝐾, if 𝐽 is already compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
kgencn3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))

Proof of Theorem kgencn3
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . . . 7 𝐽 = 𝐽
2 eqid 2732 . . . . . . 7 𝐾 = 𝐾
31, 2cnf 22741 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓: 𝐽 𝐾)
43adantl 482 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓: 𝐽 𝐾)
5 cnvimass 6077 . . . . . . . . 9 (𝑓𝑥) ⊆ dom 𝑓
64fdmd 6725 . . . . . . . . . 10 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → dom 𝑓 = 𝐽)
76adantr 481 . . . . . . . . 9 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → dom 𝑓 = 𝐽)
85, 7sseqtrid 4033 . . . . . . . 8 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ⊆ 𝐽)
9 cnvresima 6226 . . . . . . . . . . . 12 ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) = ((𝑓 “ (𝑥 ∩ (𝑓𝑦))) ∩ 𝑦)
104ad2antrr 724 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑓: 𝐽 𝐾)
11 ffun 6717 . . . . . . . . . . . . . . 15 (𝑓: 𝐽 𝐾 → Fun 𝑓)
12 inpreima 7062 . . . . . . . . . . . . . . 15 (Fun 𝑓 → (𝑓 “ (𝑥 ∩ (𝑓𝑦))) = ((𝑓𝑥) ∩ (𝑓 “ (𝑓𝑦))))
1310, 11, 123syl 18 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑓 “ (𝑥 ∩ (𝑓𝑦))) = ((𝑓𝑥) ∩ (𝑓 “ (𝑓𝑦))))
1413ineq1d 4210 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓 “ (𝑥 ∩ (𝑓𝑦))) ∩ 𝑦) = (((𝑓𝑥) ∩ (𝑓 “ (𝑓𝑦))) ∩ 𝑦))
15 in32 4220 . . . . . . . . . . . . . 14 (((𝑓𝑥) ∩ (𝑓 “ (𝑓𝑦))) ∩ 𝑦) = (((𝑓𝑥) ∩ 𝑦) ∩ (𝑓 “ (𝑓𝑦)))
16 ssrin 4232 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥) ⊆ dom 𝑓 → ((𝑓𝑥) ∩ 𝑦) ⊆ (dom 𝑓𝑦))
175, 16ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∩ 𝑦) ⊆ (dom 𝑓𝑦)
18 dminss 6149 . . . . . . . . . . . . . . . . 17 (dom 𝑓𝑦) ⊆ (𝑓 “ (𝑓𝑦))
1917, 18sstri 3990 . . . . . . . . . . . . . . . 16 ((𝑓𝑥) ∩ 𝑦) ⊆ (𝑓 “ (𝑓𝑦))
2019a1i 11 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑥) ∩ 𝑦) ⊆ (𝑓 “ (𝑓𝑦)))
21 df-ss 3964 . . . . . . . . . . . . . . 15 (((𝑓𝑥) ∩ 𝑦) ⊆ (𝑓 “ (𝑓𝑦)) ↔ (((𝑓𝑥) ∩ 𝑦) ∩ (𝑓 “ (𝑓𝑦))) = ((𝑓𝑥) ∩ 𝑦))
2220, 21sylib 217 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (((𝑓𝑥) ∩ 𝑦) ∩ (𝑓 “ (𝑓𝑦))) = ((𝑓𝑥) ∩ 𝑦))
2315, 22eqtrid 2784 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (((𝑓𝑥) ∩ (𝑓 “ (𝑓𝑦))) ∩ 𝑦) = ((𝑓𝑥) ∩ 𝑦))
2414, 23eqtrd 2772 . . . . . . . . . . . 12 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓 “ (𝑥 ∩ (𝑓𝑦))) ∩ 𝑦) = ((𝑓𝑥) ∩ 𝑦))
259, 24eqtrid 2784 . . . . . . . . . . 11 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) = ((𝑓𝑥) ∩ 𝑦))
26 simpr 485 . . . . . . . . . . . . . . 15 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
2726ad2antrr 724 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑓 ∈ (𝐽 Cn 𝐾))
28 elpwi 4608 . . . . . . . . . . . . . . 15 (𝑦 ∈ 𝒫 𝐽𝑦 𝐽)
2928ad2antrl 726 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑦 𝐽)
301cnrest 22780 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 𝐽) → (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾))
3127, 29, 30syl2anc 584 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾))
32 simpr 485 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
3332ad3antrrr 728 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝐾 ∈ Top)
34 toptopon2 22411 . . . . . . . . . . . . . . 15 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3533, 34sylib 217 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝐾 ∈ (TopOn‘ 𝐾))
36 df-ima 5688 . . . . . . . . . . . . . . . 16 (𝑓𝑦) = ran (𝑓𝑦)
3736eqimss2i 4042 . . . . . . . . . . . . . . 15 ran (𝑓𝑦) ⊆ (𝑓𝑦)
3837a1i 11 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ran (𝑓𝑦) ⊆ (𝑓𝑦))
39 imassrn 6068 . . . . . . . . . . . . . . 15 (𝑓𝑦) ⊆ ran 𝑓
4010frnd 6722 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ran 𝑓 𝐾)
4139, 40sstrid 3992 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑓𝑦) ⊆ 𝐾)
42 cnrest2 22781 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran (𝑓𝑦) ⊆ (𝑓𝑦) ∧ (𝑓𝑦) ⊆ 𝐾) → ((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾) ↔ (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦)))))
4335, 38, 41, 42syl3anc 1371 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾) ↔ (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦)))))
4431, 43mpbid 231 . . . . . . . . . . . 12 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦))))
45 simplr 767 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐾))
46 simprr 771 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝐽t 𝑦) ∈ Comp)
47 imacmp 22892 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝑦) ∈ Comp) → (𝐾t (𝑓𝑦)) ∈ Comp)
4827, 46, 47syl2anc 584 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝐾t (𝑓𝑦)) ∈ Comp)
49 kgeni 23032 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑘Gen‘𝐾) ∧ (𝐾t (𝑓𝑦)) ∈ Comp) → (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦)))
5045, 48, 49syl2anc 584 . . . . . . . . . . . 12 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦)))
51 cnima 22760 . . . . . . . . . . . 12 (((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦))) ∧ (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦))) → ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) ∈ (𝐽t 𝑦))
5244, 50, 51syl2anc 584 . . . . . . . . . . 11 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) ∈ (𝐽t 𝑦))
5325, 52eqeltrrd 2834 . . . . . . . . . 10 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦))
5453expr 457 . . . . . . . . 9 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ 𝑦 ∈ 𝒫 𝐽) → ((𝐽t 𝑦) ∈ Comp → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦)))
5554ralrimiva 3146 . . . . . . . 8 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦)))
56 kgentop 23037 . . . . . . . . . . 11 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
5756ad3antrrr 728 . . . . . . . . . 10 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ Top)
58 toptopon2 22411 . . . . . . . . . 10 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
5957, 58sylib 217 . . . . . . . . 9 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
60 elkgen 23031 . . . . . . . . 9 (𝐽 ∈ (TopOn‘ 𝐽) → ((𝑓𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((𝑓𝑥) ⊆ 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦)))))
6159, 60syl 17 . . . . . . . 8 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → ((𝑓𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((𝑓𝑥) ⊆ 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦)))))
628, 55, 61mpbir2and 711 . . . . . . 7 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ∈ (𝑘Gen‘𝐽))
63 kgenidm 23042 . . . . . . . 8 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
6463ad3antrrr 728 . . . . . . 7 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑘Gen‘𝐽) = 𝐽)
6562, 64eleqtrd 2835 . . . . . 6 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ∈ 𝐽)
6665ralrimiva 3146 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)
6756, 58sylib 217 . . . . . . 7 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ (TopOn‘ 𝐽))
68 kgentopon 23033 . . . . . . . 8 (𝐾 ∈ (TopOn‘ 𝐾) → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
6934, 68sylbi 216 . . . . . . 7 (𝐾 ∈ Top → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
70 iscn 22730 . . . . . . 7 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓: 𝐽 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)))
7167, 69, 70syl2an 596 . . . . . 6 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓: 𝐽 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)))
7271adantr 481 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓: 𝐽 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)))
734, 66, 72mpbir2and 711 . . . 4 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)))
7473ex 413 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾))))
7574ssrdv 3987 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn (𝑘Gen‘𝐾)))
7669adantl 482 . . . 4 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
77 toponcom 22421 . . . 4 ((𝐾 ∈ Top ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾)) → 𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)))
7832, 76, 77syl2anc 584 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)))
79 kgenss 23038 . . . 4 (𝐾 ∈ Top → 𝐾 ⊆ (𝑘Gen‘𝐾))
8079adantl 482 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ⊆ (𝑘Gen‘𝐾))
81 eqid 2732 . . . 4 (𝑘Gen‘𝐾) = (𝑘Gen‘𝐾)
8281cnss2 22772 . . 3 ((𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)) ∧ 𝐾 ⊆ (𝑘Gen‘𝐾)) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
8378, 80, 82syl2anc 584 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
8475, 83eqssd 3998 1 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  cin 3946  wss 3947  𝒫 cpw 4601   cuni 4907  ccnv 5674  dom cdm 5675  ran crn 5676  cres 5677  cima 5678  Fun wfun 6534  wf 6536  cfv 6540  (class class class)co 7405  t crest 17362  Topctop 22386  TopOnctopon 22403   Cn ccn 22719  Compccmp 22881  𝑘Genckgen 23028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-fin 8939  df-fi 9402  df-rest 17364  df-topgen 17385  df-top 22387  df-topon 22404  df-bases 22440  df-cn 22722  df-cmp 22882  df-kgen 23029
This theorem is referenced by:  kgen2cn  23054  txkgen  23147  qtopkgen  23205
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