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Theorem kgencn3 22154
 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 2824 . . . . . . 7 𝐽 = 𝐽
2 eqid 2824 . . . . . . 7 𝐾 = 𝐾
31, 2cnf 21842 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓: 𝐽 𝐾)
43adantl 485 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓: 𝐽 𝐾)
5 cnvimass 5932 . . . . . . . . 9 (𝑓𝑥) ⊆ dom 𝑓
64fdmd 6506 . . . . . . . . . 10 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → dom 𝑓 = 𝐽)
76adantr 484 . . . . . . . . 9 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → dom 𝑓 = 𝐽)
85, 7sseqtrid 4003 . . . . . . . 8 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ⊆ 𝐽)
9 cnvresima 6070 . . . . . . . . . . . 12 ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) = ((𝑓 “ (𝑥 ∩ (𝑓𝑦))) ∩ 𝑦)
104ad2antrr 725 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑓: 𝐽 𝐾)
11 ffun 6500 . . . . . . . . . . . . . . 15 (𝑓: 𝐽 𝐾 → Fun 𝑓)
12 inpreima 6817 . . . . . . . . . . . . . . 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 4171 . . . . . . . . . . . . 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 4181 . . . . . . . . . . . . . 14 (((𝑓𝑥) ∩ (𝑓 “ (𝑓𝑦))) ∩ 𝑦) = (((𝑓𝑥) ∩ 𝑦) ∩ (𝑓 “ (𝑓𝑦)))
16 ssrin 4193 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥) ⊆ dom 𝑓 → ((𝑓𝑥) ∩ 𝑦) ⊆ (dom 𝑓𝑦))
175, 16ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∩ 𝑦) ⊆ (dom 𝑓𝑦)
18 dminss 5993 . . . . . . . . . . . . . . . . 17 (dom 𝑓𝑦) ⊆ (𝑓 “ (𝑓𝑦))
1917, 18sstri 3960 . . . . . . . . . . . . . . . 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 3935 . . . . . . . . . . . . . . 15 (((𝑓𝑥) ∩ 𝑦) ⊆ (𝑓 “ (𝑓𝑦)) ↔ (((𝑓𝑥) ∩ 𝑦) ∩ (𝑓 “ (𝑓𝑦))) = ((𝑓𝑥) ∩ 𝑦))
2220, 21sylib 221 . . . . . . . . . . . . . 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, 22syl5eq 2871 . . . . . . . . . . . . 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 2859 . . . . . . . . . . . 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, 24syl5eq 2871 . . . . . . . . . . 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 488 . . . . . . . . . . . . . . 15 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
2726ad2antrr 725 . . . . . . . . . . . . . 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 4529 . . . . . . . . . . . . . . 15 (𝑦 ∈ 𝒫 𝐽𝑦 𝐽)
2928ad2antrl 727 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑦 𝐽)
301cnrest 21881 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 𝐽) → (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾))
3127, 29, 30syl2anc 587 . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
3332ad3antrrr 729 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝐾 ∈ Top)
34 toptopon2 21514 . . . . . . . . . . . . . . 15 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3533, 34sylib 221 . . . . . . . . . . . . . 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 5551 . . . . . . . . . . . . . . . 16 (𝑓𝑦) = ran (𝑓𝑦)
3736eqimss2i 4010 . . . . . . . . . . . . . . 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 5923 . . . . . . . . . . . . . . 15 (𝑓𝑦) ⊆ ran 𝑓
4010frnd 6504 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ran 𝑓 𝐾)
4139, 40sstrid 3962 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑓𝑦) ⊆ 𝐾)
42 cnrest2 21882 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran (𝑓𝑦) ⊆ (𝑓𝑦) ∧ (𝑓𝑦) ⊆ 𝐾) → ((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾) ↔ (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦)))))
4335, 38, 41, 42syl3anc 1368 . . . . . . . . . . . . 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 235 . . . . . . . . . . . 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 768 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐾))
46 simprr 772 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝐽t 𝑦) ∈ Comp)
47 imacmp 21993 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝑦) ∈ Comp) → (𝐾t (𝑓𝑦)) ∈ Comp)
4827, 46, 47syl2anc 587 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝐾t (𝑓𝑦)) ∈ Comp)
49 kgeni 22133 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑘Gen‘𝐾) ∧ (𝐾t (𝑓𝑦)) ∈ Comp) → (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦)))
5045, 48, 49syl2anc 587 . . . . . . . . . . . 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 21861 . . . . . . . . . . . 12 (((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦))) ∧ (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦))) → ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) ∈ (𝐽t 𝑦))
5244, 50, 51syl2anc 587 . . . . . . . . . . 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 2917 . . . . . . . . . 10 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦))
5453expr 460 . . . . . . . . 9 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ 𝑦 ∈ 𝒫 𝐽) → ((𝐽t 𝑦) ∈ Comp → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦)))
5554ralrimiva 3176 . . . . . . . 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 22138 . . . . . . . . . . 11 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
5756ad3antrrr 729 . . . . . . . . . 10 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ Top)
58 toptopon2 21514 . . . . . . . . . 10 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
5957, 58sylib 221 . . . . . . . . 9 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
60 elkgen 22132 . . . . . . . . 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 712 . . . . . . 7 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ∈ (𝑘Gen‘𝐽))
63 kgenidm 22143 . . . . . . . 8 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
6463ad3antrrr 729 . . . . . . 7 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑘Gen‘𝐽) = 𝐽)
6562, 64eleqtrd 2918 . . . . . 6 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ∈ 𝐽)
6665ralrimiva 3176 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)
6756, 58sylib 221 . . . . . . 7 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ (TopOn‘ 𝐽))
68 kgentopon 22134 . . . . . . . 8 (𝐾 ∈ (TopOn‘ 𝐾) → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
6934, 68sylbi 220 . . . . . . 7 (𝐾 ∈ Top → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
70 iscn 21831 . . . . . . 7 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓: 𝐽 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)))
7167, 69, 70syl2an 598 . . . . . 6 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓: 𝐽 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)))
7271adantr 484 . . . . 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 712 . . . 4 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)))
7473ex 416 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾))))
7574ssrdv 3957 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn (𝑘Gen‘𝐾)))
7669adantl 485 . . . 4 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
77 toponcom 21524 . . . 4 ((𝐾 ∈ Top ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾)) → 𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)))
7832, 76, 77syl2anc 587 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)))
79 kgenss 22139 . . . 4 (𝐾 ∈ Top → 𝐾 ⊆ (𝑘Gen‘𝐾))
8079adantl 485 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ⊆ (𝑘Gen‘𝐾))
81 eqid 2824 . . . 4 (𝑘Gen‘𝐾) = (𝑘Gen‘𝐾)
8281cnss2 21873 . . 3 ((𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)) ∧ 𝐾 ⊆ (𝑘Gen‘𝐾)) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
8378, 80, 82syl2anc 587 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
8475, 83eqssd 3968 1 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3132   ∩ cin 3917   ⊆ wss 3918  𝒫 cpw 4520  ∪ cuni 4821  ◡ccnv 5537  dom cdm 5538  ran crn 5539   ↾ cres 5540   “ cima 5541  Fun wfun 6332  ⟶wf 6334  ‘cfv 6338  (class class class)co 7140   ↾t crest 16685  Topctop 21489  TopOnctopon 21506   Cn ccn 21820  Compccmp 21982  𝑘Genckgen 22129 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7674  df-2nd 7675  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-er 8274  df-map 8393  df-en 8495  df-dom 8496  df-fin 8498  df-fi 8861  df-rest 16687  df-topgen 16708  df-top 21490  df-topon 21507  df-bases 21542  df-cn 21823  df-cmp 21983  df-kgen 22130 This theorem is referenced by:  kgen2cn  22155  txkgen  22248  qtopkgen  22306
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