Theorem ptpjcn 23598

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 23581	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
4	3	3adant3 1129	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
5	1, 4	eqtr4id 2784	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → 𝑌 = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
6	5	mpteq1d 5247	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) = (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)))
7		pttop 23569	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
8	7	3adant3 1129	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (∏t‘𝐹) ∈ Top)
9	2, 8	eqeltrid 2829	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → 𝐽 ∈ Top)
10		ffvelcdm 7094	. . . 4 ⊢ ((𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝐹‘𝐼) ∈ Top)
11	10	3adant1 1127	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝐹‘𝐼) ∈ Top)
12		vex 3465	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
13	12	elixp 8932	. . . . . . . . 9 ⊢ (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘)))
14	13	simprbi 495	. . . . . . . 8 ⊢ (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘))
15		fveq2 6900	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → (𝑥‘𝑘) = (𝑥‘𝐼))
16		fveq2 6900	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
17	16	unieqd 4925	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝐼))
18	15, 17	eleq12d 2819	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → ((𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘) ↔ (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼)))
19	18	rspcva 3605	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘)) → (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼))
20	14, 19	sylan2 591	. . . . . . 7 ⊢ ((𝐼 ∈ 𝐴 ∧ 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼))
21	20	3ad2antl3 1184	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼))
22	21	fmpttd 7128	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)⟶∪ (𝐹‘𝐼))
23	5	feq2d 6713	. . . . 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 256	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):𝑌⟶∪ (𝐹‘𝐼))
25		eqid 2725	. . . . . . . . . . . 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 23566	. . . . . . . . . . 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 22952	. . . . . . . . . . 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 17	. . . . . . . . . 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 6727	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
30	25	ptval 23557	. . . . . . . . . . . 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 2777	. . . . . . . . . . 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 591	. . . . . . . . . 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 4020	. . . . . . . . 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 479	. . . . . . . 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 2725	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
36	25, 35	ptpjpre2 23567	. . . . . . . 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 3979	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝐼))) → (◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)
38	37	expr 455	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝐼 ∈ 𝐴) → (𝑢 ∈ (𝐹‘𝐼) → (◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽))
39	38	ralrimiv 3134	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝐼 ∈ 𝐴) → ∀𝑢 ∈ (𝐹‘𝐼)(◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)
40	39	3impa 1107	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → ∀𝑢 ∈ (𝐹‘𝐼)(◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)
41	24, 40	jca 510	. . 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 2725	. . . 4 ⊢ ∪ (𝐹‘𝐼) = ∪ (𝐹‘𝐼)
43	1, 42	iscn2 23225	. . 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 1341	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)))
45	6, 44	eqeltrd 2825	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∀wral 3050  ∃wrex 3059   ∖ cdif 3943   ⊆ wss 3946  ∪ cuni 4912   ↦ cmpt 5235  ^◡ccnv 5680   “ cima 5684   Fn wfn 6548  ⟶wf 6549  ‘cfv 6553  (class class class)co 7423  Xcixp 8925  Fincfn 8973  topGenctg 17447  ∏_tcpt 17448  Topctop 22878  TopBasesctb 22931   Cn ccn 23211

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 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745

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 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7876  df-1o 8495  df-2o 8496  df-map 8856  df-ixp 8926  df-en 8974  df-fin 8977  df-fi 9450  df-topgen 17453  df-pt 17454  df-top 22879  df-topon 22896  df-bases 22932  df-cn 23214

This theorem is referenced by:  pthaus  23625  ptrescn  23626  xkopjcn  23643  pt1hmeo  23793  ptunhmeo  23795  tmdgsum  24082  symgtgp  24093  prdstmdd  24111  prdstgpd  24112  poimir  37302  broucube  37303