Theorem kgencn2 23714

Description: A function 𝐹:𝐽⟶𝐾 from a compactly generated space is continuous iff for all compact spaces 𝑧 and continuous 𝑔:𝑧⟶𝐽, the composite 𝐹 ∘ 𝑔:𝑧⟶𝐾 is continuous. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
kgencn2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾))))

Distinct variable groups:   𝑧,𝑔,𝐹   𝑔,𝐽,𝑧   𝑔,𝐾,𝑧   𝑔,𝑋,𝑧   𝑔,𝑌,𝑧

Proof of Theorem kgencn2

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kgencn 23713	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))))
2		rncmp 23553	. . . . . . . 8 ⊢ ((𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽)) → (𝐽 ↾t ran 𝑔) ∈ Comp)
3	2	adantl 486	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (𝐽 ↾t ran 𝑔) ∈ Comp)
4		simprr 784	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn 𝐽))
5		eqid 2763	. . . . . . . . . . . 12 ⊢ ∪ 𝑧 = ∪ 𝑧
6		eqid 2763	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 23403	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝑧 Cn 𝐽) → 𝑔:∪ 𝑧⟶∪ 𝐽)
8		frn 6713	. . . . . . . . . . 11 ⊢ (𝑔:∪ 𝑧⟶∪ 𝐽 → ran 𝑔 ⊆ ∪ 𝐽)
9	4, 7, 8	3syl 19	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ ∪ 𝐽)
10		toponuni 23071	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
11	10	ad3antrrr 742	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑋 = ∪ 𝐽)
12	9, 11	sseqtrrd 3974	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ 𝑋)
13		vex 3459	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
14	13	rnex 7903	. . . . . . . . . 10 ⊢ ran 𝑔 ∈ V
15	14	elpw 4566	. . . . . . . . 9 ⊢ (ran 𝑔 ∈ 𝒫 𝑋 ↔ ran 𝑔 ⊆ 𝑋)
16	12, 15	sylibr 237	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ∈ 𝒫 𝑋)
17		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑘 = ran 𝑔 → (𝐽 ↾t 𝑘) = (𝐽 ↾t ran 𝑔))
18	17	eleq1d 2848	. . . . . . . . . 10 ⊢ (𝑘 = ran 𝑔 → ((𝐽 ↾t 𝑘) ∈ Comp ↔ (𝐽 ↾t ran 𝑔) ∈ Comp))
19		reseq2 5973	. . . . . . . . . . 11 ⊢ (𝑘 = ran 𝑔 → (𝐹 ↾ 𝑘) = (𝐹 ↾ ran 𝑔))
20	17	oveq1d 7425	. . . . . . . . . . 11 ⊢ (𝑘 = ran 𝑔 → ((𝐽 ↾t 𝑘) Cn 𝐾) = ((𝐽 ↾t ran 𝑔) Cn 𝐾))
21	19, 20	eleq12d 2857	. . . . . . . . . 10 ⊢ (𝑘 = ran 𝑔 → ((𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ (𝐹 ↾ ran 𝑔) ∈ ((𝐽 ↾t ran 𝑔) Cn 𝐾)))
22	18, 21	imbi12d 347	. . . . . . . . 9 ⊢ (𝑘 = ran 𝑔 → (((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) ↔ ((𝐽 ↾t ran 𝑔) ∈ Comp → (𝐹 ↾ ran 𝑔) ∈ ((𝐽 ↾t ran 𝑔) Cn 𝐾))))
23	22	rspcv 3577	. . . . . . . 8 ⊢ (ran 𝑔 ∈ 𝒫 𝑋 → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) → ((𝐽 ↾t ran 𝑔) ∈ Comp → (𝐹 ↾ ran 𝑔) ∈ ((𝐽 ↾t ran 𝑔) Cn 𝐾))))
24	16, 23	syl 18	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) → ((𝐽 ↾t ran 𝑔) ∈ Comp → (𝐹 ↾ ran 𝑔) ∈ ((𝐽 ↾t ran 𝑔) Cn 𝐾))))
25	3, 24	mpid 45	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) → (𝐹 ↾ ran 𝑔) ∈ ((𝐽 ↾t ran 𝑔) Cn 𝐾)))
26		simplll 786	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝐽 ∈ (TopOn‘𝑋))
27		ssidd 3960	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ ran 𝑔)
28		cnrest2 23443	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ran 𝑔 ⊆ ran 𝑔 ∧ ran 𝑔 ⊆ 𝑋) → (𝑔 ∈ (𝑧 Cn 𝐽) ↔ 𝑔 ∈ (𝑧 Cn (𝐽 ↾t ran 𝑔))))
29	26, 27, 12, 28	syl3anc 1398	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (𝑔 ∈ (𝑧 Cn 𝐽) ↔ 𝑔 ∈ (𝑧 Cn (𝐽 ↾t ran 𝑔))))
30	4, 29	mpbid 235	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn (𝐽 ↾t ran 𝑔)))
31		cnco 23423	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑧 Cn (𝐽 ↾t ran 𝑔)) ∧ (𝐹 ↾ ran 𝑔) ∈ ((𝐽 ↾t ran 𝑔) Cn 𝐾)) → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾))
32	31	ex 417	. . . . . . . 8 ⊢ (𝑔 ∈ (𝑧 Cn (𝐽 ↾t ran 𝑔)) → ((𝐹 ↾ ran 𝑔) ∈ ((𝐽 ↾t ran 𝑔) Cn 𝐾) → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
33	30, 32	syl 18	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ((𝐹 ↾ ran 𝑔) ∈ ((𝐽 ↾t ran 𝑔) Cn 𝐾) → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
34		ssid 3959	. . . . . . . . 9 ⊢ ran 𝑔 ⊆ ran 𝑔
35		cores 6250	. . . . . . . . 9 ⊢ (ran 𝑔 ⊆ ran 𝑔 → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) = (𝐹 ∘ 𝑔))
36	34, 35	ax-mp 5	. . . . . . . 8 ⊢ ((𝐹 ↾ ran 𝑔) ∘ 𝑔) = (𝐹 ∘ 𝑔)
37	36	eleq1i 2854	. . . . . . 7 ⊢ (((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾))
38	33, 37	imbitrdi 254	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ((𝐹 ↾ ran 𝑔) ∈ ((𝐽 ↾t ran 𝑔) Cn 𝐾) → (𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
39	25, 38	syld 48	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) → (𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
40	39	ralrimdvva 3220	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) → ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
41		oveq1 7417	. . . . . . . . 9 ⊢ (𝑧 = (𝐽 ↾t 𝑘) → (𝑧 Cn 𝐽) = ((𝐽 ↾t 𝑘) Cn 𝐽))
42		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑧 = (𝐽 ↾t 𝑘) → (𝑧 Cn 𝐾) = ((𝐽 ↾t 𝑘) Cn 𝐾))
43	42	eleq2d 2849	. . . . . . . . 9 ⊢ (𝑧 = (𝐽 ↾t 𝑘) → ((𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
44	41, 43	raleqbidv 3338	. . . . . . . 8 ⊢ (𝑧 = (𝐽 ↾t 𝑘) → (∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ ∀𝑔 ∈ ((𝐽 ↾t 𝑘) Cn 𝐽)(𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
45	44	rspcv 3577	. . . . . . 7 ⊢ ((𝐽 ↾t 𝑘) ∈ Comp → (∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) → ∀𝑔 ∈ ((𝐽 ↾t 𝑘) Cn 𝐽)(𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
46		elpwi 4569	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
47	46	adantl 486	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 ⊆ 𝑋)
48	47	resabs1d 6007	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) = ( I ↾ 𝑘))
49		idcn 23414	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
50	49	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
51	10	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑋 = ∪ 𝐽)
52	47, 51	sseqtrd 3973	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 ⊆ ∪ 𝐽)
53	6	cnrest 23442	. . . . . . . . . . 11 ⊢ ((( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽) ∧ 𝑘 ⊆ ∪ 𝐽) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐽))
54	50, 52, 53	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐽))
55	48, 54	eqeltrrd 2864	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐽))
56		coeq2 5844	. . . . . . . . . . 11 ⊢ (𝑔 = ( I ↾ 𝑘) → (𝐹 ∘ 𝑔) = (𝐹 ∘ ( I ↾ 𝑘)))
57	56	eleq1d 2848	. . . . . . . . . 10 ⊢ (𝑔 = ( I ↾ 𝑘) → ((𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
58	57	rspcv 3577	. . . . . . . . 9 ⊢ (( I ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐽) → (∀𝑔 ∈ ((𝐽 ↾t 𝑘) Cn 𝐽)(𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) → (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
59	55, 58	syl 18	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (∀𝑔 ∈ ((𝐽 ↾t 𝑘) Cn 𝐽)(𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) → (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
60		coires1 6266	. . . . . . . . 9 ⊢ (𝐹 ∘ ( I ↾ 𝑘)) = (𝐹 ↾ 𝑘)
61	60	eleq1i 2854	. . . . . . . 8 ⊢ ((𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))
62	59, 61	imbitrdi 254	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (∀𝑔 ∈ ((𝐽 ↾t 𝑘) Cn 𝐽)(𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
63	45, 62	syl9r 79	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽 ↾t 𝑘) ∈ Comp → (∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))))
64	63	com23 87	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) → ((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))))
65	64	ralrimdva 3165	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) → ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))))
66	40, 65	impbid 215	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) ↔ ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
67	66	pm5.32da 589	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾))))
68	1, 67	bitrd 282	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3905  𝒫 cpw 4562  ∪ cuni 4872   I cid 5555  ran crn 5662   ↾ cres 5663   ∘ ccom 5665  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↾_t crest 17468  TopOnctopon 23067   Cn ccn 23381  Compccmp 23543  𝑘Genckgen 23690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-1o 8449  df-map 8822  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9367  df-rest 17470  df-topgen 17491  df-top 23051  df-topon 23068  df-bases 23103  df-cn 23384  df-cmp 23544  df-kgen 23691

This theorem is referenced by: (None)