Theorem ptcnp 23570

Description: If every projection of a function is continuous at 𝐷, then the function itself is continuous at 𝐷 into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
ptcnp.2	⊢ 𝐾 = (∏t‘𝐹)
ptcnp.3	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
ptcnp.4	⊢ (𝜑 → 𝐼 ∈ 𝑉)
ptcnp.5	⊢ (𝜑 → 𝐹:𝐼⟶Top)
ptcnp.6	⊢ (𝜑 → 𝐷 ∈ 𝑋)
ptcnp.7	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷))

Assertion

Ref	Expression
ptcnp	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷))

Distinct variable groups:   𝑥,𝑘,𝐷   𝑘,𝐼,𝑥   𝑘,𝐽   𝜑,𝑘,𝑥   𝑘,𝐹,𝑥   𝑘,𝑉,𝑥   𝑘,𝑋,𝑥

Allowed substitution hints:   𝐴(𝑥,𝑘)   𝐽(𝑥)   𝐾(𝑥,𝑘)

Proof of Theorem ptcnp

Dummy variables 𝑓 𝑔 𝑤 𝑧 𝑎 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptcnp.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2	1	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3		ptcnp.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐼⟶Top)
4	3	ffvelcdmda 7093	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ Top)
5		toptopon2 22864	. . . . . . . . 9 ⊢ ((𝐹‘𝑘) ∈ Top ↔ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)))
6	4, 5	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)))
7		ptcnp.7	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷))
8		cnpf2 23198	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘))
9	2, 6, 7, 8	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘))
10	9	fvmptelcdm 7122	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ (𝐹‘𝑘))
11	10	an32s 650	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝐴 ∈ ∪ (𝐹‘𝑘))
12	11	ralrimiva 3135	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘))
13		ptcnp.4	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
14	13	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐼 ∈ 𝑉)
15		mptelixpg 8954	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘)))
16	14, 15	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘)))
17	12, 16	mpbird 256	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘))
18	17	fmpttd 7124	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘))
19		df-3an 1086	. . . . . . . 8 ⊢ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)))
20		ptcnp.2	. . . . . . . . . . . . 13 ⊢ 𝐾 = (∏t‘𝐹)
21		ptcnp.6	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
22		nfv 1909	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛))
23		nfv 1909	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))
24		nfcv 2891	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘𝑋
25		nfmpt1 5257	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐼 ↦ 𝐴)
26	24, 25	nfmpt 5256	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))
27		nfcv 2891	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝐷
28	26, 27	nffv 6906	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷)
29	28	nfel1 2908	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)
30	23, 29	nfan 1894	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛))
31	22, 30	nfan 1894	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))
32		simprll 777	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → 𝑔 Fn 𝐼)
33		simprlr 778	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛))
34		fveq2 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (𝑔‘𝑛) = (𝑔‘𝑘))
35		fveq2 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
36	34, 35	eleq12d 2819	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → ((𝑔‘𝑛) ∈ (𝐹‘𝑛) ↔ (𝑔‘𝑘) ∈ (𝐹‘𝑘)))
37	36	rspccva 3605	. . . . . . . . . . . . . 14 ⊢ ((∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈ (𝐹‘𝑘))
38	33, 37	sylan 578	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈ (𝐹‘𝑘))
39		simprrl 779	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → (𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)))
40	39	simpld 493	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → 𝑤 ∈ Fin)
41	39	simprd 494	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))
42	35	unieqd 4922	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑘))
43	34, 42	eqeq12d 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → ((𝑔‘𝑛) = ∪ (𝐹‘𝑛) ↔ (𝑔‘𝑘) = ∪ (𝐹‘𝑘)))
44	43	rspccva 3605	. . . . . . . . . . . . . 14 ⊢ ((∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛) ∧ 𝑘 ∈ (𝐼 ∖ 𝑤)) → (𝑔‘𝑘) = ∪ (𝐹‘𝑘))
45	41, 44	sylan 578	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) ∧ 𝑘 ∈ (𝐼 ∖ 𝑤)) → (𝑔‘𝑘) = ∪ (𝐹‘𝑘))
46		simprrr 780	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛))
47	34	cbvixpv 8934	. . . . . . . . . . . . . 14 ⊢ X𝑛 ∈ 𝐼 (𝑔‘𝑛) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)
48	46, 47	eleqtrdi 2835	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑘 ∈ 𝐼 (𝑔‘𝑘))
49	20, 1, 13, 3, 21, 7, 31, 32, 38, 40, 45, 48	ptcnplem 23569	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘)))
50	49	anassrs 466	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛))) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛))) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘)))
51	50	expr 455	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛))) ∧ (𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
52	51	rexlimdvaa 3145	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛))) → (∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘)))))
53	52	impr 453	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
54	19, 53	sylan2b 592	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
55		eleq2 2814	. . . . . . . 8 ⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))
56	47	eqeq2i 2738	. . . . . . . . . . . 12 ⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) ↔ 𝑓 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))
57	56	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → 𝑓 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))
58	57	sseq2d 4009	. . . . . . . . . 10 ⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓 ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘)))
59	58	anbi2d 628	. . . . . . . . 9 ⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ((𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓) ↔ (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
60	59	rexbidv 3168	. . . . . . . 8 ⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → (∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓) ↔ ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
61	55, 60	imbi12d 343	. . . . . . 7 ⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ((((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)) ↔ (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘)))))
62	54, 61	syl5ibrcom 246	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))) → (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓))))
63	62	expimpd 452	. . . . 5 ⊢ (𝜑 → (((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓))))
64	63	exlimdv 1928	. . . 4 ⊢ (𝜑 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓))))
65	64	alrimiv 1922	. . 3 ⊢ (𝜑 → ∀𝑓(∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓))))
66		eqeq1 2729	. . . . . 6 ⊢ (𝑎 = 𝑓 → (𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) ↔ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)))
67	66	anbi2d 628	. . . . 5 ⊢ (𝑎 = 𝑓 → (((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))))
68	67	exbidv 1916	. . . 4 ⊢ (𝑎 = 𝑓 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) ↔ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))))
69	68	ralab 3683	. . 3 ⊢ (∀𝑓 ∈ {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))} (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)) ↔ ∀𝑓(∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓))))
70	65, 69	sylibr 233	. 2 ⊢ (𝜑 → ∀𝑓 ∈ {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))} (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)))
71	3	ffnd 6724	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐼)
72		eqid 2725	. . . . . 6 ⊢ {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))} = {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))}
73	72	ptval 23518	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 Fn 𝐼) → (∏t‘𝐹) = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))}))
74	13, 71, 73	syl2anc 582	. . . 4 ⊢ (𝜑 → (∏t‘𝐹) = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))}))
75	20, 74	eqtrid 2777	. . 3 ⊢ (𝜑 → 𝐾 = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))}))
76	3	feqmptd 6966	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘)))
77	76	fveq2d 6900	. . . . 5 ⊢ (𝜑 → (∏t‘𝐹) = (∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))))
78	20, 77	eqtrid 2777	. . . 4 ⊢ (𝜑 → 𝐾 = (∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))))
79	6	ralrimiva 3135	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐼 (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)))
80		eqid 2725	. . . . . 6 ⊢ (∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))) = (∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘)))
81	80	pttopon 23544	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) → (∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))) ∈ (TopOn‘X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘)))
82	13, 79, 81	syl2anc 582	. . . 4 ⊢ (𝜑 → (∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))) ∈ (TopOn‘X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘)))
83	78, 82	eqeltrd 2825	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘)))
84	1, 75, 83, 21	tgcnp 23201	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷) ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ∧ ∀𝑓 ∈ {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))} (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)))))
85	18, 70, 84	mpbir2and 711	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∀wral 3050  ∃wrex 3059   ∖ cdif 3941   ⊆ wss 3944  ∪ cuni 4909   ↦ cmpt 5232   “ cima 5681   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  Xcixp 8916  Fincfn 8964  topGenctg 17422  ∏_tcpt 17423  Topctop 22839  TopOnctopon 22856   CnP ccnp 23173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-1o 8487  df-er 8725  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-fin 8968  df-fi 9436  df-topgen 17428  df-pt 17429  df-top 22840  df-topon 22857  df-bases 22893  df-cnp 23176

This theorem is referenced by:  ptcn  23575