Theorem kgencn2 22616

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 22615	. 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 22455	. . . . . . . 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 769	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn 𝐽))
5		eqid 2738	. . . . . . . . . . . 12 ⊢ ∪ 𝑧 = ∪ 𝑧
6		eqid 2738	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 22305	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝑧 Cn 𝐽) → 𝑔:∪ 𝑧⟶∪ 𝐽)
8		frn 6591	. . . . . . . . . . 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 21971	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
11	10	ad3antrrr 726	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑋 = ∪ 𝐽)
12	9, 11	sseqtrrd 3958	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ 𝑋)
13		vex 3426	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
14	13	rnex 7733	. . . . . . . . . 10 ⊢ ran 𝑔 ∈ V
15	14	elpw 4534	. . . . . . . . 9 ⊢ (ran 𝑔 ∈ 𝒫 𝑋 ↔ ran 𝑔 ⊆ 𝑋)
16	12, 15	sylibr 233	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ∈ 𝒫 𝑋)
17		oveq2 7263	. . . . . . . . . . 11 ⊢ (𝑘 = ran 𝑔 → (𝐽 ↾t 𝑘) = (𝐽 ↾t ran 𝑔))
18	17	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑘 = ran 𝑔 → ((𝐽 ↾t 𝑘) ∈ Comp ↔ (𝐽 ↾t ran 𝑔) ∈ Comp))
19		reseq2 5875	. . . . . . . . . . 11 ⊢ (𝑘 = ran 𝑔 → (𝐹 ↾ 𝑘) = (𝐹 ↾ ran 𝑔))
20	17	oveq1d 7270	. . . . . . . . . . 11 ⊢ (𝑘 = ran 𝑔 → ((𝐽 ↾t 𝑘) Cn 𝐾) = ((𝐽 ↾t ran 𝑔) Cn 𝐾))
21	19, 20	eleq12d 2833	. . . . . . . . . 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 3547	. . . . . . . 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 771	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝐽 ∈ (TopOn‘𝑋))
27		ssidd 3940	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ ran 𝑔)
28		cnrest2 22345	. . . . . . . . . 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 1369	. . . . . . . . 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 231	. . . . . . . 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 22325	. . . . . . . . 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 3939	. . . . . . . . 9 ⊢ ran 𝑔 ⊆ ran 𝑔
35		cores 6142	. . . . . . . . 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 2829	. . . . . . 7 ⊢ (((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾))
38	33, 37	syl6ib 250	. . . . . 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 3117	. . . 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 7262	. . . . . . . . 9 ⊢ (𝑧 = (𝐽 ↾t 𝑘) → (𝑧 Cn 𝐽) = ((𝐽 ↾t 𝑘) Cn 𝐽))
42		oveq1 7262	. . . . . . . . . 10 ⊢ (𝑧 = (𝐽 ↾t 𝑘) → (𝑧 Cn 𝐾) = ((𝐽 ↾t 𝑘) Cn 𝐾))
43	42	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑧 = (𝐽 ↾t 𝑘) → ((𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
44	41, 43	raleqbidv 3327	. . . . . . . 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 3547	. . . . . . 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 4539	. . . . . . . . . . . 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 5911	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) = ( I ↾ 𝑘))
49		idcn 22316	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
50	49	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
51	10	ad3antrrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑋 = ∪ 𝐽)
52	47, 51	sseqtrd 3957	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 ⊆ ∪ 𝐽)
53	6	cnrest 22344	. . . . . . . . . . 11 ⊢ ((( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽) ∧ 𝑘 ⊆ ∪ 𝐽) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐽))
54	50, 52, 53	syl2anc 583	. . . . . . . . . 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 2840	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐽))
56		coeq2 5756	. . . . . . . . . . 11 ⊢ (𝑔 = ( I ↾ 𝑘) → (𝐹 ∘ 𝑔) = (𝐹 ∘ ( I ↾ 𝑘)))
57	56	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑔 = ( I ↾ 𝑘) → ((𝐹 ∘ 𝑔) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))
58	57	rspcv 3547	. . . . . . . . 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 6157	. . . . . . . . 9 ⊢ (𝐹 ∘ ( I ↾ 𝑘)) = (𝐹 ↾ 𝑘)
61	60	eleq1i 2829	. . . . . . . 8 ⊢ ((𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))
62	59, 61	syl6ib 250	. . . . . . 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 3112	. . . 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 211	. . 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 578	. 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 278	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 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   I cid 5479  ran crn 5581   ↾ cres 5582   ∘ ccom 5584  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↾_t crest 17048  TopOnctopon 21967   Cn ccn 22283  Compccmp 22445  𝑘Genckgen 22592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004  df-cn 22286  df-cmp 22446  df-kgen 22593

This theorem is referenced by: (None)