Theorem kgencn3 22709

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 2738	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2738	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 22397	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓:∪ 𝐽⟶∪ 𝐾)
4	3	adantl 482	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓:∪ 𝐽⟶∪ 𝐾)
5		cnvimass 5989	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓
6	4	fdmd 6611	. . . . . . . . . 10 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → dom 𝑓 = ∪ 𝐽)
7	6	adantr 481	. . . . . . . . 9 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → dom 𝑓 = ∪ 𝐽)
8	5, 7	sseqtrid 3973	. . . . . . . 8 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ⊆ ∪ 𝐽)
9		cnvresima 6133	. . . . . . . . . . . 12 ⊢ (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦)
10	4	ad2antrr 723	. . . . . . . . . . . . . . 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 6603	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐽⟶∪ 𝐾 → Fun 𝑓)
12		inpreima 6941	. . . . . . . . . . . . . . 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 4145	. . . . . . . . . . . . 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 4155	. . . . . . . . . . . . . 14 ⊢ (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦) = (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦)))
16		ssrin 4167	. . . . . . . . . . . . . . . . . 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 6056	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝑓 ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦))
19	17, 18	sstri 3930	. . . . . . . . . . . . . . . 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		df-ss 3904	. . . . . . . . . . . . . . 15 ⊢ (((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦)) ↔ (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
22	20, 21	sylib 217	. . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 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 2790	. . . . . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
27	26	ad2antrr 723	. . . . . . . . . . . . . 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 4542	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐽 → 𝑦 ⊆ ∪ 𝐽)
29	28	ad2antrl 725	. . . . . . . . . . . . . 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 22436	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
33	32	ad3antrrr 727	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝐾 ∈ Top)
34		toptopon2 22067	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
35	33, 34	sylib 217	. . . . . . . . . . . . . 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 5602	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 “ 𝑦) = ran (𝑓 ↾ 𝑦)
37	36	eqimss2i 3980	. . . . . . . . . . . . . . 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 5980	. . . . . . . . . . . . . . 15 ⊢ (𝑓 “ 𝑦) ⊆ ran 𝑓
40	10	frnd 6608	. . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . 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 22437	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 766	. . . . . . . . . . . . 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 770	. . . . . . . . . . . . . 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 22548	. . . . . . . . . . . . . 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 22688	. . . . . . . . . . . . 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 22416	. . . . . . . . . . . 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 2840	. . . . . . . . . 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 457	. . . . . . . . 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 3103	. . . . . . . 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 22693	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
57	56	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ Top)
58		toptopon2 22067	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
59	57, 58	sylib 217	. . . . . . . . 9 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
60		elkgen 22687	. . . . . . . . 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 710	. . . . . . 7 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽))
63		kgenidm 22698	. . . . . . . 8 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
64	63	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑘Gen‘𝐽) = 𝐽)
65	62, 64	eleqtrd 2841	. . . . . 6 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ 𝐽)
66	65	ralrimiva 3103	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ (𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽)
67	56, 58	sylib 217	. . . . . . 7 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ (TopOn‘∪ 𝐽))
68		kgentopon 22689	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
69	34, 68	sylbi 216	. . . . . . 7 ⊢ (𝐾 ∈ Top → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
70		iscn 22386	. . . . . . 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 481	. . . . 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 710	. . . 4 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)))
74	73	ex 413	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾))))
75	74	ssrdv 3927	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn (𝑘Gen‘𝐾)))
76	69	adantl 482	. . . 4 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
77		toponcom 22077	. . . 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 22694	. . . 4 ⊢ (𝐾 ∈ Top → 𝐾 ⊆ (𝑘Gen‘𝐾))
80	79	adantl 482	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ⊆ (𝑘Gen‘𝐾))
81		eqid 2738	. . . 4 ⊢ ∪ (𝑘Gen‘𝐾) = ∪ (𝑘Gen‘𝐾)
82	81	cnss2 22428	. . 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 3938	1 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591   “ cima 5592  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↾_t crest 17131  Topctop 22042  TopOnctopon 22059   Cn ccn 22375  Compccmp 22537  𝑘Genckgen 22684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-cmp 22538  df-kgen 22685

This theorem is referenced by:  kgen2cn  22710  txkgen  22803  qtopkgen  22861