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Theorem ptcnp 22234
 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 = (∏t𝐹)
ptcnp.3 (𝜑𝐽 ∈ (TopOn‘𝑋))
ptcnp.4 (𝜑𝐼𝑉)
ptcnp.5 (𝜑𝐹:𝐼⟶Top)
ptcnp.6 (𝜑𝐷𝑋)
ptcnp.7 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))
Assertion
Ref Expression
ptcnp (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((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.
StepHypRef Expression
1 ptcnp.3 . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘𝑋))
21adantr 484 . . . . . . . 8 ((𝜑𝑘𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3 ptcnp.5 . . . . . . . . . 10 (𝜑𝐹:𝐼⟶Top)
43ffvelrnda 6828 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
5 toptopon2 21530 . . . . . . . . 9 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
64, 5sylib 221 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
7 ptcnp.7 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))
8 cnpf2 21862 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷)) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
92, 6, 7, 8syl3anc 1368 . . . . . . 7 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
109fvmptelrn 6854 . . . . . 6 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
1110an32s 651 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → 𝐴 (𝐹𝑘))
1211ralrimiva 3149 . . . 4 ((𝜑𝑥𝑋) → ∀𝑘𝐼 𝐴 (𝐹𝑘))
13 ptcnp.4 . . . . . 6 (𝜑𝐼𝑉)
1413adantr 484 . . . . 5 ((𝜑𝑥𝑋) → 𝐼𝑉)
15 mptelixpg 8484 . . . . 5 (𝐼𝑉 → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
1614, 15syl 17 . . . 4 ((𝜑𝑥𝑋) → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
1712, 16mpbird 260 . . 3 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘))
1817fmpttd 6856 . 2 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋X𝑘𝐼 (𝐹𝑘))
19 df-3an 1086 . . . . . . . 8 ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)))
20 ptcnp.2 . . . . . . . . . . . . 13 𝐾 = (∏t𝐹)
21 ptcnp.6 . . . . . . . . . . . . 13 (𝜑𝐷𝑋)
22 nfv 1915 . . . . . . . . . . . . . 14 𝑘(𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))
23 nfv 1915 . . . . . . . . . . . . . . 15 𝑘(𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))
24 nfcv 2955 . . . . . . . . . . . . . . . . . 18 𝑘𝑋
25 nfmpt1 5128 . . . . . . . . . . . . . . . . . 18 𝑘(𝑘𝐼𝐴)
2624, 25nfmpt 5127 . . . . . . . . . . . . . . . . 17 𝑘(𝑥𝑋 ↦ (𝑘𝐼𝐴))
27 nfcv 2955 . . . . . . . . . . . . . . . . 17 𝑘𝐷
2826, 27nffv 6655 . . . . . . . . . . . . . . . 16 𝑘((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
2928nfel1 2971 . . . . . . . . . . . . . . 15 𝑘((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)
3023, 29nfan 1900 . . . . . . . . . . . . . 14 𝑘((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛))
3122, 30nfan 1900 . . . . . . . . . . . . 13 𝑘((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))
32 simprll 778 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → 𝑔 Fn 𝐼)
33 simprlr 779 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))
34 fveq2 6645 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑔𝑛) = (𝑔𝑘))
35 fveq2 6645 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
3634, 35eleq12d 2884 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑔𝑛) ∈ (𝐹𝑛) ↔ (𝑔𝑘) ∈ (𝐹𝑘)))
3736rspccva 3570 . . . . . . . . . . . . . 14 ((∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ 𝑘𝐼) → (𝑔𝑘) ∈ (𝐹𝑘))
3833, 37sylan 583 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) ∧ 𝑘𝐼) → (𝑔𝑘) ∈ (𝐹𝑘))
39 simprrl 780 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → (𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)))
4039simpld 498 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → 𝑤 ∈ Fin)
4139simprd 499 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))
4235unieqd 4814 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
4334, 42eqeq12d 2814 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑔𝑛) = (𝐹𝑛) ↔ (𝑔𝑘) = (𝐹𝑘)))
4443rspccva 3570 . . . . . . . . . . . . . 14 ((∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛) ∧ 𝑘 ∈ (𝐼𝑤)) → (𝑔𝑘) = (𝐹𝑘))
4541, 44sylan 583 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) ∧ 𝑘 ∈ (𝐼𝑤)) → (𝑔𝑘) = (𝐹𝑘))
46 simprrr 781 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛))
4734cbvixpv 8464 . . . . . . . . . . . . . 14 X𝑛𝐼 (𝑔𝑛) = X𝑘𝐼 (𝑔𝑘)
4846, 47eleqtrdi 2900 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑘𝐼 (𝑔𝑘))
4920, 1, 13, 3, 21, 7, 31, 32, 38, 40, 45, 48ptcnplem 22233 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘)))
5049anassrs 471 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘)))
5150expr 460 . . . . . . . . . 10 (((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))) ∧ (𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘))))
5251rexlimdvaa 3244 . . . . . . . . 9 ((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))) → (∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘)))))
5352impr 458 . . . . . . . 8 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘))))
5419, 53sylan2b 596 . . . . . . 7 ((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘))))
55 eleq2 2878 . . . . . . . 8 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))
5647eqeq2i 2811 . . . . . . . . . . . 12 (𝑓 = X𝑛𝐼 (𝑔𝑛) ↔ 𝑓 = X𝑘𝐼 (𝑔𝑘))
5756biimpi 219 . . . . . . . . . . 11 (𝑓 = X𝑛𝐼 (𝑔𝑛) → 𝑓 = X𝑘𝐼 (𝑔𝑘))
5857sseq2d 3947 . . . . . . . . . 10 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓 ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘)))
5958anbi2d 631 . . . . . . . . 9 (𝑓 = X𝑛𝐼 (𝑔𝑛) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓) ↔ (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘))))
6059rexbidv 3256 . . . . . . . 8 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓) ↔ ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘))))
6155, 60imbi12d 348 . . . . . . 7 (𝑓 = X𝑛𝐼 (𝑔𝑛) → ((((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓)) ↔ (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘)))))
6254, 61syl5ibrcom 250 . . . . . 6 ((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))) → (𝑓 = X𝑛𝐼 (𝑔𝑛) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓))))
6362expimpd 457 . . . . 5 (𝜑 → (((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑓 = X𝑛𝐼 (𝑔𝑛)) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓))))
6463exlimdv 1934 . . . 4 (𝜑 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑓 = X𝑛𝐼 (𝑔𝑛)) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓))))
6564alrimiv 1928 . . 3 (𝜑 → ∀𝑓(∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑓 = X𝑛𝐼 (𝑔𝑛)) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓))))
66 eqeq1 2802 . . . . . 6 (𝑎 = 𝑓 → (𝑎 = X𝑛𝐼 (𝑔𝑛) ↔ 𝑓 = X𝑛𝐼 (𝑔𝑛)))
6766anbi2d 631 . . . . 5 (𝑎 = 𝑓 → (((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑓 = X𝑛𝐼 (𝑔𝑛))))
6867exbidv 1922 . . . 4 (𝑎 = 𝑓 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛)) ↔ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑓 = X𝑛𝐼 (𝑔𝑛))))
6968ralab 3632 . . 3 (∀𝑓 ∈ {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))} (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓)) ↔ ∀𝑓(∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑓 = X𝑛𝐼 (𝑔𝑛)) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓))))
7065, 69sylibr 237 . 2 (𝜑 → ∀𝑓 ∈ {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))} (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓)))
713ffnd 6488 . . . . 5 (𝜑𝐹 Fn 𝐼)
72 eqid 2798 . . . . . 6 {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))} = {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))}
7372ptval 22182 . . . . 5 ((𝐼𝑉𝐹 Fn 𝐼) → (∏t𝐹) = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))}))
7413, 71, 73syl2anc 587 . . . 4 (𝜑 → (∏t𝐹) = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))}))
7520, 74syl5eq 2845 . . 3 (𝜑𝐾 = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))}))
763feqmptd 6708 . . . . . 6 (𝜑𝐹 = (𝑘𝐼 ↦ (𝐹𝑘)))
7776fveq2d 6649 . . . . 5 (𝜑 → (∏t𝐹) = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))))
7820, 77syl5eq 2845 . . . 4 (𝜑𝐾 = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))))
796ralrimiva 3149 . . . . 5 (𝜑 → ∀𝑘𝐼 (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
80 eqid 2798 . . . . . 6 (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘)))
8180pttopon 22208 . . . . 5 ((𝐼𝑉 ∧ ∀𝑘𝐼 (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘))) → (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) ∈ (TopOn‘X𝑘𝐼 (𝐹𝑘)))
8213, 79, 81syl2anc 587 . . . 4 (𝜑 → (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) ∈ (TopOn‘X𝑘𝐼 (𝐹𝑘)))
8378, 82eqeltrd 2890 . . 3 (𝜑𝐾 ∈ (TopOn‘X𝑘𝐼 (𝐹𝑘)))
841, 75, 83, 21tgcnp 21865 . 2 (𝜑 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋X𝑘𝐼 (𝐹𝑘) ∧ ∀𝑓 ∈ {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))} (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓)))))
8518, 70, 84mpbir2and 712 1 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ⊆ wss 3881  ∪ cuni 4800   ↦ cmpt 5110   “ cima 5522   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Xcixp 8446  Fincfn 8494  topGenctg 16705  ∏tcpt 16706  Topctop 21505  TopOnctopon 21522   CnP ccnp 21837 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-er 8274  df-map 8393  df-ixp 8447  df-en 8495  df-dom 8496  df-fin 8498  df-fi 8861  df-topgen 16711  df-pt 16712  df-top 21506  df-topon 21523  df-bases 21558  df-cnp 21840 This theorem is referenced by:  ptcn  22239
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