Theorem ptpjcn 23737

Description: Continuity of a projection map into a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Feb-2015.)

Hypotheses

Ref	Expression
ptpjcn.1	⊢ 𝑌 = ∪ 𝐽
ptpjcn.2	⊢ 𝐽 = (∏t‘𝐹)

Assertion

Ref	Expression
ptpjcn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐼   𝑥,𝑉   𝑥,𝑌

Allowed substitution hint:   𝐽(𝑥)

Proof of Theorem ptpjcn

Dummy variables 𝑔 𝑘 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptpjcn.1	. . . 4 ⊢ 𝑌 = ∪ 𝐽
2		ptpjcn.2	. . . . . 6 ⊢ 𝐽 = (∏t‘𝐹)
3	2	ptuni 23720	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
4	3	3adant3 1148	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
5	1, 4	eqtr4id 2823	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → 𝑌 = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
6	5	mpteq1d 5205	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) = (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)))
7		pttop 23708	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
8	7	3adant3 1148	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (∏t‘𝐹) ∈ Top)
9	2, 8	eqeltrid 2873	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → 𝐽 ∈ Top)
10		ffvelcdm 7077	. . . 4 ⊢ ((𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝐹‘𝐼) ∈ Top)
11	10	3adant1 1146	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝐹‘𝐼) ∈ Top)
12		vex 3467	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
13	12	elixp 8902	. . . . . . . . 9 ⊢ (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘)))
14	13	simprbi 502	. . . . . . . 8 ⊢ (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘))
15		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → (𝑥‘𝑘) = (𝑥‘𝐼))
16		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
17	16	unieqd 4889	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝐼))
18	15, 17	eleq12d 2863	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → ((𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘) ↔ (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼)))
19	18	rspcva 3588	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘)) → (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼))
20	14, 19	sylan2 604	. . . . . . 7 ⊢ ((𝐼 ∈ 𝐴 ∧ 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼))
21	20	3ad2antl3 1204	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼))
22	21	fmpttd 7111	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)⟶∪ (𝐹‘𝐼))
23	5	feq2d 6690	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → ((𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):𝑌⟶∪ (𝐹‘𝐼) ↔ (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)⟶∪ (𝐹‘𝐼)))
24	22, 23	mpbird 260	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):𝑌⟶∪ (𝐹‘𝐼))
25		eqid 2769	. . . . . . . . . . . 12 ⊢ {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}
26	25	ptbas 23705	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ∈ TopBases)
27		bastg 23092	. . . . . . . . . . 11 ⊢ ({𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ∈ TopBases → {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ⊆ (topGen‘{𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
28	26, 27	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ⊆ (topGen‘{𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
29		ffn 6706	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
30	25	ptval 23696	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (∏t‘𝐹) = (topGen‘{𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
31	2, 30	eqtrid 2816	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → 𝐽 = (topGen‘{𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
32	29, 31	sylan2 604	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
33	28, 32	sseqtrrd 3982	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ⊆ 𝐽)
34	33	adantr 485	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝐼))) → {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ⊆ 𝐽)
35		eqid 2769	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
36	25, 35	ptpjpre2 23706	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝐼))) → (◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))})
37	34, 36	sseldd 3946	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝐼))) → (◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)
38	37	expr 461	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝐼 ∈ 𝐴) → (𝑢 ∈ (𝐹‘𝐼) → (◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽))
39	38	ralrimiv 3162	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝐼 ∈ 𝐴) → ∀𝑢 ∈ (𝐹‘𝐼)(◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)
40	39	3impa 1125	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → ∀𝑢 ∈ (𝐹‘𝐼)(◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)
41	24, 40	jca 520	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → ((𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):𝑌⟶∪ (𝐹‘𝐼) ∧ ∀𝑢 ∈ (𝐹‘𝐼)(◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽))
42		eqid 2769	. . . 4 ⊢ ∪ (𝐹‘𝐼) = ∪ (𝐹‘𝐼)
43	1, 42	iscn2 23364	. . 3 ⊢ ((𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)) ↔ ((𝐽 ∈ Top ∧ (𝐹‘𝐼) ∈ Top) ∧ ((𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):𝑌⟶∪ (𝐹‘𝐼) ∧ ∀𝑢 ∈ (𝐹‘𝐼)(◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)))
44	9, 11, 41, 43	syl21anbrc 1361	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)))
45	6, 44	eqeltrd 2869	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910   ⊆ wss 3913  ∪ cuni 4876   ↦ cmpt 5196  ^◡ccnv 5661   “ cima 5665   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  Xcixp 8895  Fincfn 8943  topGenctg 17490  ∏_tcpt 17491  Topctop 23019  TopBasesctb 23071   Cn ccn 23350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1o 8453  df-2o 8454  df-map 8826  df-ixp 8896  df-en 8944  df-fin 8947  df-fi 9371  df-topgen 17496  df-pt 17497  df-top 23020  df-topon 23037  df-bases 23072  df-cn 23353

This theorem is referenced by:  pthaus  23764  ptrescn  23765  xkopjcn  23782  pt1hmeo  23932  ptunhmeo  23934  tmdgsum  24221  symgtgp  24232  prdstmdd  24250  prdstgpd  24251  poimir  38226  broucube  38227