Theorem kgencn3 23494

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 2735	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2735	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 23182	. . . . . 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 6069	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓
6	4	fdmd 6715	. . . . . . . . . 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 4001	. . . . . . . 8 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ⊆ ∪ 𝐽)
9		cnvresima 6219	. . . . . . . . . . . 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 6708	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐽⟶∪ 𝐾 → Fun 𝑓)
12		inpreima 7053	. . . . . . . . . . . . . . 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 4194	. . . . . . . . . . . . 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 4205	. . . . . . . . . . . . . 14 ⊢ (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦) = (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦)))
16		ssrin 4217	. . . . . . . . . . . . . . . . . 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 6142	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝑓 ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦))
19	17, 18	sstri 3968	. . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . 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 2770	. . . . . . . . . . . 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 2782	. . . . . . . . . . 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 4582	. . . . . . . . . . . . . . 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 23221	. . . . . . . . . . . . . 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 22854	. . . . . . . . . . . . . . 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 5667	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 “ 𝑦) = ran (𝑓 ↾ 𝑦)
37	36	eqimss2i 4020	. . . . . . . . . . . . . . 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 6058	. . . . . . . . . . . . . . 15 ⊢ (𝑓 “ 𝑦) ⊆ ran 𝑓
40	10	frnd 6713	. . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . 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 23222	. . . . . . . . . . . . . 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 23333	. . . . . . . . . . . . . 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 23473	. . . . . . . . . . . . 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 23201	. . . . . . . . . . . 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 2835	. . . . . . . . . 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 3132	. . . . . . . 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 23478	. . . . . . . . . . 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 22854	. . . . . . . . . 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 23472	. . . . . . . . 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 23483	. . . . . . . 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 2836	. . . . . 6 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ 𝐽)
66	65	ralrimiva 3132	. . . . 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 23474	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
69	34, 68	sylbi 217	. . . . . . 7 ⊢ (𝐾 ∈ Top → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
70		iscn 23171	. . . . . . 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 3964	. 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 22864	. . . 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 23479	. . . 4 ⊢ (𝐾 ∈ Top → 𝐾 ⊆ (𝑘Gen‘𝐾))
80	79	adantl 481	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ⊆ (𝑘Gen‘𝐾))
81		eqid 2735	. . . 4 ⊢ ∪ (𝑘Gen‘𝐾) = ∪ (𝑘Gen‘𝐾)
82	81	cnss2 23213	. . 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 3976	1 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575  ∪ cuni 4883  ^◡ccnv 5653  dom cdm 5654  ran crn 5655   ↾ cres 5656   “ cima 5657  Fun wfun 6524  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403   ↾_t crest 17432  Topctop 22829  TopOnctopon 22846   Cn ccn 23160  Compccmp 23322  𝑘Genckgen 23469

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-1o 8478  df-map 8840  df-en 8958  df-dom 8959  df-fin 8961  df-fi 9421  df-rest 17434  df-topgen 17455  df-top 22830  df-topon 22847  df-bases 22882  df-cn 23163  df-cmp 23323  df-kgen 23470

This theorem is referenced by:  kgen2cn  23495  txkgen  23588  qtopkgen  23646