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Theorem kgencn3 21266
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 2626 . . . . . . 7 𝐽 = 𝐽
2 eqid 2626 . . . . . . 7 𝐾 = 𝐾
31, 2cnf 20955 . . . . . 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 5448 . . . . . . . . 9 (𝑓𝑥) ⊆ dom 𝑓
6 fdm 6010 . . . . . . . . . . 11 (𝑓: 𝐽 𝐾 → dom 𝑓 = 𝐽)
74, 6syl 17 . . . . . . . . . 10 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → dom 𝑓 = 𝐽)
87adantr 481 . . . . . . . . 9 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → dom 𝑓 = 𝐽)
95, 8syl5sseq 3637 . . . . . . . 8 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ⊆ 𝐽)
10 cnvresima 5585 . . . . . . . . . . . 12 ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) = ((𝑓 “ (𝑥 ∩ (𝑓𝑦))) ∩ 𝑦)
114ad2antrr 761 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑓: 𝐽 𝐾)
12 ffun 6007 . . . . . . . . . . . . . . 15 (𝑓: 𝐽 𝐾 → Fun 𝑓)
13 inpreima 6299 . . . . . . . . . . . . . . 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 3796 . . . . . . . . . . . . 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 3808 . . . . . . . . . . . . . 14 (((𝑓𝑥) ∩ (𝑓 “ (𝑓𝑦))) ∩ 𝑦) = (((𝑓𝑥) ∩ 𝑦) ∩ (𝑓 “ (𝑓𝑦)))
17 ssrin 3821 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥) ⊆ dom 𝑓 → ((𝑓𝑥) ∩ 𝑦) ⊆ (dom 𝑓𝑦))
185, 17ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∩ 𝑦) ⊆ (dom 𝑓𝑦)
19 dminss 5510 . . . . . . . . . . . . . . . . 17 (dom 𝑓𝑦) ⊆ (𝑓 “ (𝑓𝑦))
2018, 19sstri 3597 . . . . . . . . . . . . . . . 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 3574 . . . . . . . . . . . . . . 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 2672 . . . . . . . . . . . . 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 2660 . . . . . . . . . . . 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 2672 . . . . . . . . . . 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 477 . . . . . . . . . . . . . . 15 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
2827ad2antrr 761 . . . . . . . . . . . . . 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 4145 . . . . . . . . . . . . . . 15 (𝑦 ∈ 𝒫 𝐽𝑦 𝐽)
3029ad2antrl 763 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑦 𝐽)
311cnrest 20994 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 𝐽) → (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾))
3228, 30, 31syl2anc 692 . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
3433ad3antrrr 765 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝐾 ∈ Top)
352toptopon 20643 . . . . . . . . . . . . . . 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 5092 . . . . . . . . . . . . . . . 16 (𝑓𝑦) = ran (𝑓𝑦)
3837eqimss2i 3644 . . . . . . . . . . . . . . 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 5440 . . . . . . . . . . . . . . 15 (𝑓𝑦) ⊆ ran 𝑓
41 frn 6012 . . . . . . . . . . . . . . . 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 3599 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑓𝑦) ⊆ 𝐾)
44 cnrest2 20995 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran (𝑓𝑦) ⊆ (𝑓𝑦) ∧ (𝑓𝑦) ⊆ 𝐾) → ((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾) ↔ (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦)))))
4536, 39, 43, 44syl3anc 1323 . . . . . . . . . . . . 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 791 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐾))
48 simprr 795 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝐽t 𝑦) ∈ Comp)
49 imacmp 21105 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝑦) ∈ Comp) → (𝐾t (𝑓𝑦)) ∈ Comp)
5028, 48, 49syl2anc 692 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝐾t (𝑓𝑦)) ∈ Comp)
51 kgeni 21245 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑘Gen‘𝐾) ∧ (𝐾t (𝑓𝑦)) ∈ Comp) → (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦)))
5247, 50, 51syl2anc 692 . . . . . . . . . . . 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 20974 . . . . . . . . . . . 12 (((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦))) ∧ (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦))) → ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) ∈ (𝐽t 𝑦))
5446, 52, 53syl2anc 692 . . . . . . . . . . 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 2705 . . . . . . . . . 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 2965 . . . . . . . 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 21250 . . . . . . . . . . 11 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
5958ad3antrrr 765 . . . . . . . . . 10 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ Top)
601toptopon 20643 . . . . . . . . . 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 21244 . . . . . . . . 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 956 . . . . . . 7 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ∈ (𝑘Gen‘𝐽))
65 kgenidm 21255 . . . . . . . 8 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
6665ad3antrrr 765 . . . . . . 7 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑘Gen‘𝐽) = 𝐽)
6764, 66eleqtrd 2706 . . . . . 6 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ∈ 𝐽)
6867ralrimiva 2965 . . . . 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 21246 . . . . . . . 8 (𝐾 ∈ (TopOn‘ 𝐾) → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
7135, 70sylbi 207 . . . . . . 7 (𝐾 ∈ Top → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
72 iscn 20944 . . . . . . 7 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓: 𝐽 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)))
7369, 71, 72syl2an 494 . . . . . 6 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓: 𝐽 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)))
7473adantr 481 . . . . 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 956 . . . 4 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)))
7675ex 450 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾))))
7776ssrdv 3594 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn (𝑘Gen‘𝐾)))
7871adantl 482 . . . 4 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
79 toponcom 20640 . . . 4 ((𝐾 ∈ Top ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾)) → 𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)))
8033, 78, 79syl2anc 692 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)))
81 kgenss 21251 . . . 4 (𝐾 ∈ Top → 𝐾 ⊆ (𝑘Gen‘𝐾))
8281adantl 482 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ⊆ (𝑘Gen‘𝐾))
83 eqid 2626 . . . 4 (𝑘Gen‘𝐾) = (𝑘Gen‘𝐾)
8483cnss2 20986 . . 3 ((𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)) ∧ 𝐾 ⊆ (𝑘Gen‘𝐾)) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
8580, 82, 84syl2anc 692 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
8677, 85eqssd 3605 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 384   = wceq 1480  wcel 1992  wral 2912  cin 3559  wss 3560  𝒫 cpw 4135   cuni 4407  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  Fun wfun 5844  wf 5846  cfv 5850  (class class class)co 6605  t crest 15997  Topctop 20612  TopOnctopon 20613   Cn ccn 20933  Compccmp 21094  𝑘Genckgen 21241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-fin 7904  df-fi 8262  df-rest 15999  df-topgen 16020  df-top 20616  df-bases 20617  df-topon 20618  df-cn 20936  df-cmp 21095  df-kgen 21242
This theorem is referenced by:  kgen2cn  21267  txkgen  21360  qtopkgen  21418
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