Theorem kgencn3 22946

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 2731	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2731	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 22634	. . . . . 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 6038	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓
6	4	fdmd 6684	. . . . . . . . . 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 3999	. . . . . . . 8 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ⊆ ∪ 𝐽)
9		cnvresima 6187	. . . . . . . . . . . 12 ⊢ (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦)
10	4	ad2antrr 724	. . . . . . . . . . . . . . 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 6676	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐽⟶∪ 𝐾 → Fun 𝑓)
12		inpreima 7019	. . . . . . . . . . . . . . 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 4176	. . . . . . . . . . . . 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 4186	. . . . . . . . . . . . . 14 ⊢ (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦) = (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦)))
16		ssrin 4198	. . . . . . . . . . . . . . . . . 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 6110	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝑓 ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦))
19	17, 18	sstri 3956	. . . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 2783	. . . . . . . . . . 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 724	. . . . . . . . . . . . . 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 4572	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐽 → 𝑦 ⊆ ∪ 𝐽)
29	28	ad2antrl 726	. . . . . . . . . . . . . 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 22673	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝐾 ∈ Top)
34		toptopon2 22304	. . . . . . . . . . . . . . 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 5651	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 “ 𝑦) = ran (𝑓 ↾ 𝑦)
37	36	eqimss2i 4008	. . . . . . . . . . . . . . 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 6029	. . . . . . . . . . . . . . 15 ⊢ (𝑓 “ 𝑦) ⊆ ran 𝑓
40	10	frnd 6681	. . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . 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 22674	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 767	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 22785	. . . . . . . . . . . . . 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 22925	. . . . . . . . . . . . 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 22653	. . . . . . . . . . . 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 2833	. . . . . . . . . 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 3139	. . . . . . . 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 22930	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
57	56	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ Top)
58		toptopon2 22304	. . . . . . . . . 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 22924	. . . . . . . . 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 711	. . . . . . 7 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽))
63		kgenidm 22935	. . . . . . . 8 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
64	63	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑘Gen‘𝐽) = 𝐽)
65	62, 64	eleqtrd 2834	. . . . . 6 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ 𝐽)
66	65	ralrimiva 3139	. . . . 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 22926	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
69	34, 68	sylbi 216	. . . . . . 7 ⊢ (𝐾 ∈ Top → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
70		iscn 22623	. . . . . . 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 711	. . . 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 3953	. 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 22314	. . . 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 22931	. . . 4 ⊢ (𝐾 ∈ Top → 𝐾 ⊆ (𝑘Gen‘𝐾))
80	79	adantl 482	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ⊆ (𝑘Gen‘𝐾))
81		eqid 2731	. . . 4 ⊢ ∪ (𝑘Gen‘𝐾) = ∪ (𝑘Gen‘𝐾)
82	81	cnss2 22665	. . 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 3964	1 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4565  ∪ cuni 4870  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641  Fun wfun 6495  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362   ↾_t crest 17316  Topctop 22279  TopOnctopon 22296   Cn ccn 22612  Compccmp 22774  𝑘Genckgen 22921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-fin 8894  df-fi 9356  df-rest 17318  df-topgen 17339  df-top 22280  df-topon 22297  df-bases 22333  df-cn 22615  df-cmp 22775  df-kgen 22922

This theorem is referenced by:  kgen2cn  22947  txkgen  23040  qtopkgen  23098