Theorem kgencn2 23420

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 23419	. 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 23259	. . . . . . . 8 ⊢ ((𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽)) → (𝐽 ↾t ran 𝑔) ∈ Comp)
3	2	adantl 481	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (𝐽 ↾t ran 𝑔) ∈ Comp)
4		simprr 772	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn 𝐽))
5		eqid 2729	. . . . . . . . . . . 12 ⊢ ∪ 𝑧 = ∪ 𝑧
6		eqid 2729	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 23109	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝑧 Cn 𝐽) → 𝑔:∪ 𝑧⟶∪ 𝐽)
8		frn 6677	. . . . . . . . . . 11 ⊢ (𝑔:∪ 𝑧⟶∪ 𝐽 → ran 𝑔 ⊆ ∪ 𝐽)
9	4, 7, 8	3syl 18	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ ∪ 𝐽)
10		toponuni 22777	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
11	10	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑋 = ∪ 𝐽)
12	9, 11	sseqtrrd 3981	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ 𝑋)
13		vex 3448	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
14	13	rnex 7866	. . . . . . . . . 10 ⊢ ran 𝑔 ∈ V
15	14	elpw 4563	. . . . . . . . 9 ⊢ (ran 𝑔 ∈ 𝒫 𝑋 ↔ ran 𝑔 ⊆ 𝑋)
16	12, 15	sylibr 234	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ∈ 𝒫 𝑋)
17		oveq2 7377	. . . . . . . . . . 11 ⊢ (𝑘 = ran 𝑔 → (𝐽 ↾t 𝑘) = (𝐽 ↾t ran 𝑔))
18	17	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑘 = ran 𝑔 → ((𝐽 ↾t 𝑘) ∈ Comp ↔ (𝐽 ↾t ran 𝑔) ∈ Comp))
19		reseq2 5934	. . . . . . . . . . 11 ⊢ (𝑘 = ran 𝑔 → (𝐹 ↾ 𝑘) = (𝐹 ↾ ran 𝑔))
20	17	oveq1d 7384	. . . . . . . . . . 11 ⊢ (𝑘 = ran 𝑔 → ((𝐽 ↾t 𝑘) Cn 𝐾) = ((𝐽 ↾t ran 𝑔) Cn 𝐾))
21	19, 20	eleq12d 2822	. . . . . . . . . 10 ⊢ (𝑘 = ran 𝑔 → ((𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ (𝐹 ↾ ran 𝑔) ∈ ((𝐽 ↾t ran 𝑔) Cn 𝐾)))
22	18, 21	imbi12d 344	. . . . . . . . 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 3581	. . . . . . . 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 17	. . . . . . 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 44	. . . . . 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 774	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝐽 ∈ (TopOn‘𝑋))
27		ssidd 3967	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ ran 𝑔)
28		cnrest2 23149	. . . . . . . . . 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 1373	. . . . . . . . 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 232	. . . . . . . 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 23129	. . . . . . . . 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 412	. . . . . . . 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 17	. . . . . . 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 3966	. . . . . . . . 9 ⊢ ran 𝑔 ⊆ ran 𝑔
35		cores 6210	. . . . . . . . 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 2819	. . . . . . 7 ⊢ (((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾))
38	33, 37	imbitrdi 251	. . . . . 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 47	. . . . 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 3190	. . . 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 7376	. . . . . . . . 9 ⊢ (𝑧 = (𝐽 ↾t 𝑘) → (𝑧 Cn 𝐽) = ((𝐽 ↾t 𝑘) Cn 𝐽))
42		oveq1 7376	. . . . . . . . . 10 ⊢ (𝑧 = (𝐽 ↾t 𝑘) → (𝑧 Cn 𝐾) = ((𝐽 ↾t 𝑘) Cn 𝐾))
43	42	eleq2d 2814	. . . . . . . . 9 ⊢ (𝑧 = (𝐽 ↾t 𝑘) → ((𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
44	41, 43	raleqbidv 3316	. . . . . . . 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 3581	. . . . . . 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 4566	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
47	46	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 ⊆ 𝑋)
48	47	resabs1d 5968	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) = ( I ↾ 𝑘))
49		idcn 23120	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
50	49	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
51	10	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑋 = ∪ 𝐽)
52	47, 51	sseqtrd 3980	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 ⊆ ∪ 𝐽)
53	6	cnrest 23148	. . . . . . . . . . 11 ⊢ ((( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽) ∧ 𝑘 ⊆ ∪ 𝐽) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐽))
54	50, 52, 53	syl2anc 584	. . . . . . . . . 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 2829	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐽))
56		coeq2 5812	. . . . . . . . . . 11 ⊢ (𝑔 = ( I ↾ 𝑘) → (𝐹 ∘ 𝑔) = (𝐹 ∘ ( I ↾ 𝑘)))
57	56	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑔 = ( I ↾ 𝑘) → ((𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
58	57	rspcv 3581	. . . . . . . . 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 17	. . . . . . . 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 6225	. . . . . . . . 9 ⊢ (𝐹 ∘ ( I ↾ 𝑘)) = (𝐹 ↾ 𝑘)
61	60	eleq1i 2819	. . . . . . . 8 ⊢ ((𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))
62	59, 61	imbitrdi 251	. . . . . . 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 78	. . . . . 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 86	. . . . 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 3133	. . . 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 212	. . 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 579	. 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 279	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867   I cid 5525  ran crn 5632   ↾ cres 5633   ∘ ccom 5635  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ↾_t crest 17359  TopOnctopon 22773   Cn ccn 23087  Compccmp 23249  𝑘Genckgen 23396

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-1o 8411  df-map 8778  df-en 8896  df-dom 8897  df-fin 8899  df-fi 9338  df-rest 17361  df-topgen 17382  df-top 22757  df-topon 22774  df-bases 22809  df-cn 23090  df-cmp 23250  df-kgen 23397

This theorem is referenced by: (None)