Theorem ptpjcn 23498

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 23481	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
4	3	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
5	1, 4	eqtr4id 2783	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → 𝑌 = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
6	5	mpteq1d 5197	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) = (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)))
7		pttop 23469	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
8	7	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (∏t‘𝐹) ∈ Top)
9	2, 8	eqeltrid 2832	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → 𝐽 ∈ Top)
10		ffvelcdm 7053	. . . 4 ⊢ ((𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝐹‘𝐼) ∈ Top)
11	10	3adant1 1130	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝐹‘𝐼) ∈ Top)
12		vex 3451	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
13	12	elixp 8877	. . . . . . . . 9 ⊢ (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘)))
14	13	simprbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 (𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘))
15		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → (𝑥‘𝑘) = (𝑥‘𝐼))
16		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
17	16	unieqd 4884	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝐼))
18	15, 17	eleq12d 2822	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → ((𝑥‘𝑘) ∈ ∪ (𝐹‘𝑘) ↔ (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼)))
19	18	rspcva 3586	. . . . . . . 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 1188	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑥‘𝐼) ∈ ∪ (𝐹‘𝐼))
22	21	fmpttd 7087	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)⟶∪ (𝐹‘𝐼))
23	5	feq2d 6672	. . . . 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 257	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)):𝑌⟶∪ (𝐹‘𝐼))
25		eqid 2729	. . . . . . . . . . . 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 23466	. . . . . . . . . . 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 22853	. . . . . . . . . . 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 6688	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
30	25	ptval 23457	. . . . . . . . . . . 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 2776	. . . . . . . . . . 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 3984	. . . . . . . . 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 480	. . . . . . . 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 2729	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
36	25, 35	ptpjpre2 23467	. . . . . . . 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 3947	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝐼))) → (◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽)
38	37	expr 456	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝐼 ∈ 𝐴) → (𝑢 ∈ (𝐹‘𝐼) → (◡(𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) “ 𝑢) ∈ 𝐽))
39	38	ralrimiv 3124	. . . . 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 511	. . 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 2729	. . . 4 ⊢ ∪ (𝐹‘𝐼) = ∪ (𝐹‘𝐼)
43	1, 42	iscn2 23125	. . 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 1345	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)))
45	6, 44	eqeltrd 2828	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ∖ cdif 3911   ⊆ wss 3914  ∪ cuni 4871   ↦ cmpt 5188  ^◡ccnv 5637   “ cima 5641   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Xcixp 8870  Fincfn 8918  topGenctg 17400  ∏_tcpt 17401  Topctop 22780  TopBasesctb 22832   Cn ccn 23111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1o 8434  df-2o 8435  df-map 8801  df-ixp 8871  df-en 8919  df-fin 8922  df-fi 9362  df-topgen 17406  df-pt 17407  df-top 22781  df-topon 22798  df-bases 22833  df-cn 23114

This theorem is referenced by:  pthaus  23525  ptrescn  23526  xkopjcn  23543  pt1hmeo  23693  ptunhmeo  23695  tmdgsum  23982  symgtgp  23993  prdstmdd  24011  prdstgpd  24012  poimir  37647  broucube  37648