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Theorem kgencn3 21409
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 2651 . . . . . . 7 𝐽 = 𝐽
2 eqid 2651 . . . . . . 7 𝐾 = 𝐾
31, 2cnf 21098 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓: 𝐽 𝐾)
43adantl 481 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓: 𝐽 𝐾)
5 cnvimass 5520 . . . . . . . . 9 (𝑓𝑥) ⊆ dom 𝑓
6 fdm 6089 . . . . . . . . . . 11 (𝑓: 𝐽 𝐾 → dom 𝑓 = 𝐽)
74, 6syl 17 . . . . . . . . . 10 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → dom 𝑓 = 𝐽)
87adantr 480 . . . . . . . . 9 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → dom 𝑓 = 𝐽)
95, 8syl5sseq 3686 . . . . . . . 8 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ⊆ 𝐽)
10 cnvresima 5661 . . . . . . . . . . . 12 ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) = ((𝑓 “ (𝑥 ∩ (𝑓𝑦))) ∩ 𝑦)
114ad2antrr 762 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑓: 𝐽 𝐾)
12 ffun 6086 . . . . . . . . . . . . . . 15 (𝑓: 𝐽 𝐾 → Fun 𝑓)
13 inpreima 6382 . . . . . . . . . . . . . . 15 (Fun 𝑓 → (𝑓 “ (𝑥 ∩ (𝑓𝑦))) = ((𝑓𝑥) ∩ (𝑓 “ (𝑓𝑦))))
1411, 12, 133syl 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))))
1514ineq1d 3846 . . . . . . . . . . . . 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))
16 in32 3858 . . . . . . . . . . . . . 14 (((𝑓𝑥) ∩ (𝑓 “ (𝑓𝑦))) ∩ 𝑦) = (((𝑓𝑥) ∩ 𝑦) ∩ (𝑓 “ (𝑓𝑦)))
17 ssrin 3871 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥) ⊆ dom 𝑓 → ((𝑓𝑥) ∩ 𝑦) ⊆ (dom 𝑓𝑦))
185, 17ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∩ 𝑦) ⊆ (dom 𝑓𝑦)
19 dminss 5582 . . . . . . . . . . . . . . . . 17 (dom 𝑓𝑦) ⊆ (𝑓 “ (𝑓𝑦))
2018, 19sstri 3645 . . . . . . . . . . . . . . . 16 ((𝑓𝑥) ∩ 𝑦) ⊆ (𝑓 “ (𝑓𝑦))
2120a1i 11 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑥) ∩ 𝑦) ⊆ (𝑓 “ (𝑓𝑦)))
22 df-ss 3621 . . . . . . . . . . . . . . 15 (((𝑓𝑥) ∩ 𝑦) ⊆ (𝑓 “ (𝑓𝑦)) ↔ (((𝑓𝑥) ∩ 𝑦) ∩ (𝑓 “ (𝑓𝑦))) = ((𝑓𝑥) ∩ 𝑦))
2321, 22sylib 208 . . . . . . . . . . . . . 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))
2416, 23syl5eq 2697 . . . . . . . . . . . . 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))
2515, 24eqtrd 2685 . . . . . . . . . . . 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))
2610, 25syl5eq 2697 . . . . . . . . . . 11 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) = ((𝑓𝑥) ∩ 𝑦))
27 simpr 476 . . . . . . . . . . . . . . 15 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
2827ad2antrr 762 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑓 ∈ (𝐽 Cn 𝐾))
29 elpwi 4201 . . . . . . . . . . . . . . 15 (𝑦 ∈ 𝒫 𝐽𝑦 𝐽)
3029ad2antrl 764 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑦 𝐽)
311cnrest 21137 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 𝐽) → (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾))
3228, 30, 31syl2anc 694 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾))
33 simpr 476 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
3433ad3antrrr 766 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝐾 ∈ Top)
352toptopon 20770 . . . . . . . . . . . . . . 15 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3634, 35sylib 208 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝐾 ∈ (TopOn‘ 𝐾))
37 df-ima 5156 . . . . . . . . . . . . . . . 16 (𝑓𝑦) = ran (𝑓𝑦)
3837eqimss2i 3693 . . . . . . . . . . . . . . 15 ran (𝑓𝑦) ⊆ (𝑓𝑦)
3938a1i 11 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ran (𝑓𝑦) ⊆ (𝑓𝑦))
40 imassrn 5512 . . . . . . . . . . . . . . 15 (𝑓𝑦) ⊆ ran 𝑓
41 frn 6091 . . . . . . . . . . . . . . . 16 (𝑓: 𝐽 𝐾 → ran 𝑓 𝐾)
4211, 41syl 17 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ran 𝑓 𝐾)
4340, 42syl5ss 3647 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑓𝑦) ⊆ 𝐾)
44 cnrest2 21138 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran (𝑓𝑦) ⊆ (𝑓𝑦) ∧ (𝑓𝑦) ⊆ 𝐾) → ((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾) ↔ (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦)))))
4536, 39, 43, 44syl3anc 1366 . . . . . . . . . . . . 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)))))
4632, 45mpbid 222 . . . . . . . . . . . 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))))
47 simplr 807 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐾))
48 simprr 811 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝐽t 𝑦) ∈ Comp)
49 imacmp 21248 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝑦) ∈ Comp) → (𝐾t (𝑓𝑦)) ∈ Comp)
5028, 48, 49syl2anc 694 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝐾t (𝑓𝑦)) ∈ Comp)
51 kgeni 21388 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑘Gen‘𝐾) ∧ (𝐾t (𝑓𝑦)) ∈ Comp) → (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦)))
5247, 50, 51syl2anc 694 . . . . . . . . . . . 12 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦)))
53 cnima 21117 . . . . . . . . . . . 12 (((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦))) ∧ (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦))) → ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) ∈ (𝐽t 𝑦))
5446, 52, 53syl2anc 694 . . . . . . . . . . 11 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) ∈ (𝐽t 𝑦))
5526, 54eqeltrrd 2731 . . . . . . . . . 10 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦))
5655expr 642 . . . . . . . . 9 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ 𝑦 ∈ 𝒫 𝐽) → ((𝐽t 𝑦) ∈ Comp → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦)))
5756ralrimiva 2995 . . . . . . . 8 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦)))
58 kgentop 21393 . . . . . . . . . . 11 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
5958ad3antrrr 766 . . . . . . . . . 10 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ Top)
601toptopon 20770 . . . . . . . . . 10 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
6159, 60sylib 208 . . . . . . . . 9 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
62 elkgen 21387 . . . . . . . . 9 (𝐽 ∈ (TopOn‘ 𝐽) → ((𝑓𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((𝑓𝑥) ⊆ 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦)))))
6361, 62syl 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)))))
649, 57, 63mpbir2and 977 . . . . . . 7 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ∈ (𝑘Gen‘𝐽))
65 kgenidm 21398 . . . . . . . 8 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
6665ad3antrrr 766 . . . . . . 7 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑘Gen‘𝐽) = 𝐽)
6764, 66eleqtrd 2732 . . . . . 6 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ∈ 𝐽)
6867ralrimiva 2995 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)
6958, 60sylib 208 . . . . . . 7 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ (TopOn‘ 𝐽))
70 kgentopon 21389 . . . . . . . 8 (𝐾 ∈ (TopOn‘ 𝐾) → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
7135, 70sylbi 207 . . . . . . 7 (𝐾 ∈ Top → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
72 iscn 21087 . . . . . . 7 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓: 𝐽 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)))
7369, 71, 72syl2an 493 . . . . . 6 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓: 𝐽 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)))
7473adantr 480 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓: 𝐽 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)))
754, 68, 74mpbir2and 977 . . . 4 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)))
7675ex 449 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾))))
7776ssrdv 3642 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn (𝑘Gen‘𝐾)))
7871adantl 481 . . . 4 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
79 toponcom 20780 . . . 4 ((𝐾 ∈ Top ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾)) → 𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)))
8033, 78, 79syl2anc 694 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)))
81 kgenss 21394 . . . 4 (𝐾 ∈ Top → 𝐾 ⊆ (𝑘Gen‘𝐾))
8281adantl 481 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ⊆ (𝑘Gen‘𝐾))
83 eqid 2651 . . . 4 (𝑘Gen‘𝐾) = (𝑘Gen‘𝐾)
8483cnss2 21129 . . 3 ((𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)) ∧ 𝐾 ⊆ (𝑘Gen‘𝐾)) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
8580, 82, 84syl2anc 694 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
8677, 85eqssd 3653 1 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Fun wfun 5920  wf 5922  cfv 5926  (class class class)co 6690  t crest 16128  Topctop 20746  TopOnctopon 20763   Cn ccn 21076  Compccmp 21237  𝑘Genckgen 21384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079  df-cmp 21238  df-kgen 21385
This theorem is referenced by:  kgen2cn  21410  txkgen  21503  qtopkgen  21561
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