Theorem kgencn3 23443

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.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2729	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 23131	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓:∪ 𝐽⟶∪ 𝐾)
4	3	adantl 481	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓:∪ 𝐽⟶∪ 𝐾)
5		cnvimass 6033	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓
6	4	fdmd 6662	. . . . . . . . . 10 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → dom 𝑓 = ∪ 𝐽)
7	6	adantr 480	. . . . . . . . 9 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → dom 𝑓 = ∪ 𝐽)
8	5, 7	sseqtrid 3978	. . . . . . . 8 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ⊆ ∪ 𝐽)
9		cnvresima 6179	. . . . . . . . . . . 12 ⊢ (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦)
10	4	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝑓:∪ 𝐽⟶∪ 𝐾)
11		ffun 6655	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐽⟶∪ 𝐾 → Fun 𝑓)
12		inpreima 6998	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝑓 → (◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))))
13	10, 11, 12	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))))
14	13	ineq1d 4170	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦) = (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦))
15		in32 4181	. . . . . . . . . . . . . 14 ⊢ (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦) = (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦)))
16		ssrin 4193	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝑓 “ 𝑥) ⊆ dom 𝑓 → ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (dom 𝑓 ∩ 𝑦))
17	5, 16	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (dom 𝑓 ∩ 𝑦)
18		dminss 6102	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝑓 ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦))
19	17, 18	sstri 3945	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦))
20	19	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦)))
21		dfss2 3921	. . . . . . . . . . . . . . 15 ⊢ (((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦)) ↔ (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
22	20, 21	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
23	15, 22	eqtrid 2776	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
24	14, 23	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
25	9, 24	eqtrid 2776	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
26		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
27	26	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝑓 ∈ (𝐽 Cn 𝐾))
28		elpwi 4558	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐽 → 𝑦 ⊆ ∪ 𝐽)
29	28	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝑦 ⊆ ∪ 𝐽)
30	1	cnrest 23170	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ⊆ ∪ 𝐽) → (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾))
31	27, 29, 30	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾))
32		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
33	32	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝐾 ∈ Top)
34		toptopon2 22803	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
35	33, 34	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
36		df-ima 5632	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 “ 𝑦) = ran (𝑓 ↾ 𝑦)
37	36	eqimss2i 3997	. . . . . . . . . . . . . . 15 ⊢ ran (𝑓 ↾ 𝑦) ⊆ (𝑓 “ 𝑦)
38	37	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ran (𝑓 ↾ 𝑦) ⊆ (𝑓 “ 𝑦))
39		imassrn 6022	. . . . . . . . . . . . . . 15 ⊢ (𝑓 “ 𝑦) ⊆ ran 𝑓
40	10	frnd 6660	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ran 𝑓 ⊆ ∪ 𝐾)
41	39, 40	sstrid 3947	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝑓 “ 𝑦) ⊆ ∪ 𝐾)
42		cnrest2 23171	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ran (𝑓 ↾ 𝑦) ⊆ (𝑓 “ 𝑦) ∧ (𝑓 “ 𝑦) ⊆ ∪ 𝐾) → ((𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾) ↔ (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦)))))
43	35, 38, 41, 42	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ((𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾) ↔ (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦)))))
44	31, 43	mpbid 232	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦))))
45		simplr 768	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐾))
46		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝐽 ↾t 𝑦) ∈ Comp)
47		imacmp 23282	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝑦) ∈ Comp) → (𝐾 ↾t (𝑓 “ 𝑦)) ∈ Comp)
48	27, 46, 47	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝐾 ↾t (𝑓 “ 𝑦)) ∈ Comp)
49		kgeni 23422	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑘Gen‘𝐾) ∧ (𝐾 ↾t (𝑓 “ 𝑦)) ∈ Comp) → (𝑥 ∩ (𝑓 “ 𝑦)) ∈ (𝐾 ↾t (𝑓 “ 𝑦)))
50	45, 48, 49	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝑥 ∩ (𝑓 “ 𝑦)) ∈ (𝐾 ↾t (𝑓 “ 𝑦)))
51		cnima 23150	. . . . . . . . . . . 12 ⊢ (((𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦))) ∧ (𝑥 ∩ (𝑓 “ 𝑦)) ∈ (𝐾 ↾t (𝑓 “ 𝑦))) → (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) ∈ (𝐽 ↾t 𝑦))
52	44, 50, 51	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) ∈ (𝐽 ↾t 𝑦))
53	25, 52	eqeltrrd 2829	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦))
54	53	expr 456	. . . . . . . . 9 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ 𝑦 ∈ 𝒫 ∪ 𝐽) → ((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))
55	54	ralrimiva 3121	. . . . . . . 8 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))
56		kgentop 23427	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
57	56	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ Top)
58		toptopon2 22803	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
59	57, 58	sylib 218	. . . . . . . . 9 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
60		elkgen 23421	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → ((◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((◡𝑓 “ 𝑥) ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))))
61	59, 60	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → ((◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((◡𝑓 “ 𝑥) ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))))
62	8, 55, 61	mpbir2and 713	. . . . . . 7 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽))
63		kgenidm 23432	. . . . . . . 8 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
64	63	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑘Gen‘𝐽) = 𝐽)
65	62, 64	eleqtrd 2830	. . . . . 6 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ 𝐽)
66	65	ralrimiva 3121	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ (𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽)
67	56, 58	sylib 218	. . . . . . 7 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ (TopOn‘∪ 𝐽))
68		kgentopon 23423	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
69	34, 68	sylbi 217	. . . . . . 7 ⊢ (𝐾 ∈ Top → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
70		iscn 23120	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽)))
71	67, 69, 70	syl2an 596	. . . . . 6 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽)))
72	71	adantr 480	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽)))
73	4, 66, 72	mpbir2and 713	. . . 4 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)))
74	73	ex 412	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾))))
75	74	ssrdv 3941	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn (𝑘Gen‘𝐾)))
76	69	adantl 481	. . . 4 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
77		toponcom 22813	. . . 4 ⊢ ((𝐾 ∈ Top ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾)) → 𝐾 ∈ (TopOn‘∪ (𝑘Gen‘𝐾)))
78	32, 76, 77	syl2anc 584	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘∪ (𝑘Gen‘𝐾)))
79		kgenss 23428	. . . 4 ⊢ (𝐾 ∈ Top → 𝐾 ⊆ (𝑘Gen‘𝐾))
80	79	adantl 481	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ⊆ (𝑘Gen‘𝐾))
81		eqid 2729	. . . 4 ⊢ ∪ (𝑘Gen‘𝐾) = ∪ (𝑘Gen‘𝐾)
82	81	cnss2 23162	. . 3 ⊢ ((𝐾 ∈ (TopOn‘∪ (𝑘Gen‘𝐾)) ∧ 𝐾 ⊆ (𝑘Gen‘𝐾)) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
83	78, 80, 82	syl2anc 584	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
84	75, 83	eqssd 3953	1 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622  Fun wfun 6476  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↾_t crest 17324  Topctop 22778  TopOnctopon 22795   Cn ccn 23109  Compccmp 23271  𝑘Genckgen 23418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-1o 8388  df-map 8755  df-en 8873  df-dom 8874  df-fin 8876  df-fi 9301  df-rest 17326  df-topgen 17347  df-top 22779  df-topon 22796  df-bases 22831  df-cn 23112  df-cmp 23272  df-kgen 23419

This theorem is referenced by:  kgen2cn  23444  txkgen  23537  qtopkgen  23595