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Theorem kgencn3 21171
 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 2610 . . . . . . 7 𝐽 = 𝐽
2 eqid 2610 . . . . . . 7 𝐾 = 𝐾
31, 2cnf 20860 . . . . . 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 5404 . . . . . . . . 9 (𝑓𝑥) ⊆ dom 𝑓
6 fdm 5964 . . . . . . . . . . 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 3616 . . . . . . . 8 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ⊆ 𝐽)
10 cnvresima 5541 . . . . . . . . . . . 12 ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) = ((𝑓 “ (𝑥 ∩ (𝑓𝑦))) ∩ 𝑦)
114ad2antrr 758 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑓: 𝐽 𝐾)
12 ffun 5961 . . . . . . . . . . . . . . 15 (𝑓: 𝐽 𝐾 → Fun 𝑓)
13 inpreima 6250 . . . . . . . . . . . . . . 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 3775 . . . . . . . . . . . . 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 3787 . . . . . . . . . . . . . 14 (((𝑓𝑥) ∩ (𝑓 “ (𝑓𝑦))) ∩ 𝑦) = (((𝑓𝑥) ∩ 𝑦) ∩ (𝑓 “ (𝑓𝑦)))
17 ssrin 3800 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥) ⊆ dom 𝑓 → ((𝑓𝑥) ∩ 𝑦) ⊆ (dom 𝑓𝑦))
185, 17ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∩ 𝑦) ⊆ (dom 𝑓𝑦)
19 dminss 5466 . . . . . . . . . . . . . . . . 17 (dom 𝑓𝑦) ⊆ (𝑓 “ (𝑓𝑦))
2018, 19sstri 3577 . . . . . . . . . . . . . . . 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 3554 . . . . . . . . . . . . . . 15 (((𝑓𝑥) ∩ 𝑦) ⊆ (𝑓 “ (𝑓𝑦)) ↔ (((𝑓𝑥) ∩ 𝑦) ∩ (𝑓 “ (𝑓𝑦))) = ((𝑓𝑥) ∩ 𝑦))
2321, 22sylib 207 . . . . . . . . . . . . . 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 2656 . . . . . . . . . . . . 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 2644 . . . . . . . . . . . 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 2656 . . . . . . . . . . 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 758 . . . . . . . . . . . . . 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 4117 . . . . . . . . . . . . . . 15 (𝑦 ∈ 𝒫 𝐽𝑦 𝐽)
3029ad2antrl 760 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑦 𝐽)
311cnrest 20899 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 𝐽) → (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾))
3228, 30, 31syl2anc 691 . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝐾 ∈ Top)
352toptopon 20548 . . . . . . . . . . . . . . 15 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3634, 35sylib 207 . . . . . . . . . . . . . 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 5051 . . . . . . . . . . . . . . . 16 (𝑓𝑦) = ran (𝑓𝑦)
3837eqimss2i 3623 . . . . . . . . . . . . . . 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 5396 . . . . . . . . . . . . . . 15 (𝑓𝑦) ⊆ ran 𝑓
41 frn 5966 . . . . . . . . . . . . . . . 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 3579 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝑓𝑦) ⊆ 𝐾)
44 cnrest2 20900 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran (𝑓𝑦) ⊆ (𝑓𝑦) ∧ (𝑓𝑦) ⊆ 𝐾) → ((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn 𝐾) ↔ (𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦)))))
4536, 39, 43, 44syl3anc 1318 . . . . . . . . . . . . 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 221 . . . . . . . . . . . 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 788 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐾))
48 simprr 792 . . . . . . . . . . . . . 14 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝐽t 𝑦) ∈ Comp)
49 imacmp 21010 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝑦) ∈ Comp) → (𝐾t (𝑓𝑦)) ∈ Comp)
5028, 48, 49syl2anc 691 . . . . . . . . . . . . 13 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → (𝐾t (𝑓𝑦)) ∈ Comp)
51 kgeni 21150 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑘Gen‘𝐾) ∧ (𝐾t (𝑓𝑦)) ∈ Comp) → (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦)))
5247, 50, 51syl2anc 691 . . . . . . . . . . . 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 20879 . . . . . . . . . . . 12 (((𝑓𝑦) ∈ ((𝐽t 𝑦) Cn (𝐾t (𝑓𝑦))) ∧ (𝑥 ∩ (𝑓𝑦)) ∈ (𝐾t (𝑓𝑦))) → ((𝑓𝑦) “ (𝑥 ∩ (𝑓𝑦))) ∈ (𝐽t 𝑦))
5446, 52, 53syl2anc 691 . . . . . . . . . . 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 2689 . . . . . . . . . 10 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 𝐽 ∧ (𝐽t 𝑦) ∈ Comp)) → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦))
5655expr 641 . . . . . . . . 9 (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ 𝑦 ∈ 𝒫 𝐽) → ((𝐽t 𝑦) ∈ Comp → ((𝑓𝑥) ∩ 𝑦) ∈ (𝐽t 𝑦)))
5756ralrimiva 2949 . . . . . . . 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 21155 . . . . . . . . . . 11 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
5958ad3antrrr 762 . . . . . . . . . 10 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ Top)
601toptopon 20548 . . . . . . . . . 10 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
6159, 60sylib 207 . . . . . . . . 9 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
62 elkgen 21149 . . . . . . . . 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 959 . . . . . . 7 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ∈ (𝑘Gen‘𝐽))
65 kgenidm 21160 . . . . . . . 8 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
6665ad3antrrr 762 . . . . . . 7 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑘Gen‘𝐽) = 𝐽)
6764, 66eleqtrd 2690 . . . . . 6 ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑓𝑥) ∈ 𝐽)
6867ralrimiva 2949 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ (𝑘Gen‘𝐾)(𝑓𝑥) ∈ 𝐽)
6958, 60sylib 207 . . . . . . 7 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ (TopOn‘ 𝐽))
70 kgentopon 21151 . . . . . . . 8 (𝐾 ∈ (TopOn‘ 𝐾) → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
7135, 70sylbi 206 . . . . . . 7 (𝐾 ∈ Top → (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾))
72 iscn 20849 . . . . . . 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 959 . . . 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 3574 . 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 20545 . . . 4 ((𝐾 ∈ Top ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘ 𝐾)) → 𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)))
8033, 78, 79syl2anc 691 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)))
81 kgenss 21156 . . . 4 (𝐾 ∈ Top → 𝐾 ⊆ (𝑘Gen‘𝐾))
8281adantl 481 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ⊆ (𝑘Gen‘𝐾))
83 eqid 2610 . . . 4 (𝑘Gen‘𝐾) = (𝑘Gen‘𝐾)
8483cnss2 20891 . . 3 ((𝐾 ∈ (TopOn‘ (𝑘Gen‘𝐾)) ∧ 𝐾 ⊆ (𝑘Gen‘𝐾)) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
8580, 82, 84syl2anc 691 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
8677, 85eqssd 3585 1 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  Compccmp 20999  𝑘Genckgen 21146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-cmp 21000  df-kgen 21147 This theorem is referenced by:  kgen2cn  21172  txkgen  21265  qtopkgen  21323
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