Theorem ptpjcn 22762

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 22745	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
4	3	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
5	1, 4	eqtr4id 2797	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → 𝑌 = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
6	5	mpteq1d 5169	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) = (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)))
7		pttop 22733	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
8	7	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (∏t‘𝐹) ∈ Top)
9	2, 8	eqeltrid 2843	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → 𝐽 ∈ Top)
10		ffvelrn 6959	. . . 4 ⊢ ((𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝐹‘𝐼) ∈ Top)
11	10	3adant1 1129	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝐹‘𝐼) ∈ Top)
12		vex 3436	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
13	12	elixp 8692	. . . . . . . . 9 ⊢ (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘)))
14	13	simprbi 497	. . . . . . . 8 ⊢ (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘))
15		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → (𝑥‘𝑘) = (𝑥‘𝐼))
16		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
17	16	unieqd 4853	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝐼))
18	15, 17	eleq12d 2833	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → ((𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘) ↔ (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼)))
19	18	rspcva 3559	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘)) → (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼))
20	14, 19	sylan2 593	. . . . . . 7 ⊢ ((𝐼 ∈ 𝐴 ∧ 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼))
21	20	3ad2antl3 1186	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼))
22	21	fmpttd 6989	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)⟶∪ (𝐹‘𝐼))
23	5	feq2d 6586	. . . . 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 2738	. . . . . . . . . . . 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 22730	. . . . . . . . . . 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 22116	. . . . . . . . . . 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 6600	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
30	25	ptval 22721	. . . . . . . . . . . 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 2790	. . . . . . . . . . 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 593	. . . . . . . . . 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 3962	. . . . . . . . 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 481	. . . . . . . 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 2738	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
36	25, 35	ptpjpre2 22731	. . . . . . . 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 3922	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝐼))) → (◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)
38	37	expr 457	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝐼 ∈ 𝐴) → (𝑢 ∈ (𝐹‘𝐼) → (◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽))
39	38	ralrimiv 3102	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝐼 ∈ 𝐴) → ∀𝑢 ∈ (𝐹‘𝐼)(◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)
40	39	3impa 1109	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → ∀𝑢 ∈ (𝐹‘𝐼)(◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)
41	24, 40	jca 512	. . 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 2738	. . . 4 ⊢ ∪ (𝐹‘𝐼) = ∪ (𝐹‘𝐼)
43	1, 42	iscn2 22389	. . 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 1343	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)))
45	6, 44	eqeltrd 2839	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884   ⊆ wss 3887  ∪ cuni 4839   ↦ cmpt 5157  ^◡ccnv 5588   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Xcixp 8685  Fincfn 8733  topGenctg 17148  ∏_tcpt 17149  Topctop 22042  TopBasesctb 22095   Cn ccn 22375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1o 8297  df-er 8498  df-map 8617  df-ixp 8686  df-en 8734  df-fin 8737  df-fi 9170  df-topgen 17154  df-pt 17155  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378

This theorem is referenced by:  pthaus  22789  ptrescn  22790  xkopjcn  22807  pt1hmeo  22957  ptunhmeo  22959  tmdgsum  23246  symgtgp  23257  prdstmdd  23275  prdstgpd  23276  poimir  35810  broucube  35811