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Theorem ptcnp 22844
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 481 . . . . . . . 8 ((𝜑𝑘𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3 ptcnp.5 . . . . . . . . . 10 (𝜑𝐹:𝐼⟶Top)
43ffvelcdmda 6998 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
5 toptopon2 22138 . . . . . . . . 9 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
64, 5sylib 217 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
7 ptcnp.7 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))
8 cnpf2 22472 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷)) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
92, 6, 7, 8syl3anc 1370 . . . . . . 7 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
109fvmptelcdm 7024 . . . . . 6 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
1110an32s 649 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → 𝐴 (𝐹𝑘))
1211ralrimiva 3140 . . . 4 ((𝜑𝑥𝑋) → ∀𝑘𝐼 𝐴 (𝐹𝑘))
13 ptcnp.4 . . . . . 6 (𝜑𝐼𝑉)
1413adantr 481 . . . . 5 ((𝜑𝑥𝑋) → 𝐼𝑉)
15 mptelixpg 8769 . . . . 5 (𝐼𝑉 → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
1614, 15syl 17 . . . 4 ((𝜑𝑥𝑋) → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
1712, 16mpbird 256 . . 3 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘))
1817fmpttd 7026 . 2 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋X𝑘𝐼 (𝐹𝑘))
19 df-3an 1088 . . . . . . . 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 1916 . . . . . . . . . . . . . 14 𝑘(𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))
23 nfv 1916 . . . . . . . . . . . . . . 15 𝑘(𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))
24 nfcv 2905 . . . . . . . . . . . . . . . . . 18 𝑘𝑋
25 nfmpt1 5193 . . . . . . . . . . . . . . . . . 18 𝑘(𝑘𝐼𝐴)
2624, 25nfmpt 5192 . . . . . . . . . . . . . . . . 17 𝑘(𝑥𝑋 ↦ (𝑘𝐼𝐴))
27 nfcv 2905 . . . . . . . . . . . . . . . . 17 𝑘𝐷
2826, 27nffv 6819 . . . . . . . . . . . . . . . 16 𝑘((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
2928nfel1 2921 . . . . . . . . . . . . . . 15 𝑘((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)
3023, 29nfan 1901 . . . . . . . . . . . . . 14 𝑘((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛))
3122, 30nfan 1901 . . . . . . . . . . . . 13 𝑘((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))
32 simprll 776 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → 𝑔 Fn 𝐼)
33 simprlr 777 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))
34 fveq2 6809 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑔𝑛) = (𝑔𝑘))
35 fveq2 6809 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
3634, 35eleq12d 2832 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑔𝑛) ∈ (𝐹𝑛) ↔ (𝑔𝑘) ∈ (𝐹𝑘)))
3736rspccva 3569 . . . . . . . . . . . . . 14 ((∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ 𝑘𝐼) → (𝑔𝑘) ∈ (𝐹𝑘))
3833, 37sylan 580 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) ∧ 𝑘𝐼) → (𝑔𝑘) ∈ (𝐹𝑘))
39 simprrl 778 . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → 𝑤 ∈ Fin)
4139simprd 496 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))
4235unieqd 4862 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
4334, 42eqeq12d 2753 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑔𝑛) = (𝐹𝑛) ↔ (𝑔𝑘) = (𝐹𝑘)))
4443rspccva 3569 . . . . . . . . . . . . . 14 ((∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛) ∧ 𝑘 ∈ (𝐼𝑤)) → (𝑔𝑘) = (𝐹𝑘))
4541, 44sylan 580 . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . 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 8749 . . . . . . . . . . . . . 14 X𝑛𝐼 (𝑔𝑛) = X𝑘𝐼 (𝑔𝑘)
4846, 47eleqtrdi 2848 . . . . . . . . . . . . 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 22843 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 457 . . . . . . . . . 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 3150 . . . . . . . . 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 455 . . . . . . . 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 594 . . . . . . 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 2826 . . . . . . . 8 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))
5647eqeq2i 2750 . . . . . . . . . . . 12 (𝑓 = X𝑛𝐼 (𝑔𝑛) ↔ 𝑓 = X𝑘𝐼 (𝑔𝑘))
5756biimpi 215 . . . . . . . . . . 11 (𝑓 = X𝑛𝐼 (𝑔𝑛) → 𝑓 = X𝑘𝐼 (𝑔𝑘))
5857sseq2d 3962 . . . . . . . . . 10 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓 ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘)))
5958anbi2d 629 . . . . . . . . 9 (𝑓 = X𝑛𝐼 (𝑔𝑛) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓) ↔ (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘))))
6059rexbidv 3172 . . . . . . . 8 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓) ↔ ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘))))
6155, 60imbi12d 344 . . . . . . 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 246 . . . . . 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 454 . . . . 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 1935 . . . 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 1929 . . 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 2741 . . . . . 6 (𝑎 = 𝑓 → (𝑎 = X𝑛𝐼 (𝑔𝑛) ↔ 𝑓 = X𝑛𝐼 (𝑔𝑛)))
6766anbi2d 629 . . . . 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 1923 . . . 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 3637 . . 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 233 . 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 6636 . . . . 5 (𝜑𝐹 Fn 𝐼)
72 eqid 2737 . . . . . 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 22792 . . . . 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 584 . . . 4 (𝜑 → (∏t𝐹) = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))}))
7520, 74eqtrid 2789 . . 3 (𝜑𝐾 = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))}))
763feqmptd 6874 . . . . . 6 (𝜑𝐹 = (𝑘𝐼 ↦ (𝐹𝑘)))
7776fveq2d 6813 . . . . 5 (𝜑 → (∏t𝐹) = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))))
7820, 77eqtrid 2789 . . . 4 (𝜑𝐾 = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))))
796ralrimiva 3140 . . . . 5 (𝜑 → ∀𝑘𝐼 (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
80 eqid 2737 . . . . . 6 (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘)))
8180pttopon 22818 . . . . 5 ((𝐼𝑉 ∧ ∀𝑘𝐼 (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘))) → (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) ∈ (TopOn‘X𝑘𝐼 (𝐹𝑘)))
8213, 79, 81syl2anc 584 . . . 4 (𝜑 → (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) ∈ (TopOn‘X𝑘𝐼 (𝐹𝑘)))
8378, 82eqeltrd 2838 . . 3 (𝜑𝐾 ∈ (TopOn‘X𝑘𝐼 (𝐹𝑘)))
841, 75, 83, 21tgcnp 22475 . 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 710 1 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086  wal 1538   = wceq 1540  wex 1780  wcel 2105  {cab 2714  wral 3062  wrex 3071  cdif 3893  wss 3896   cuni 4848  cmpt 5168  cima 5608   Fn wfn 6458  wf 6459  cfv 6463  (class class class)co 7313  Xcixp 8731  Fincfn 8779  topGenctg 17215  tcpt 17216  Topctop 22113  TopOnctopon 22130   CnP ccnp 22447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-int 4891  df-iun 4937  df-iin 4938  df-br 5086  df-opab 5148  df-mpt 5169  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-ord 6289  df-on 6290  df-lim 6291  df-suc 6292  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-om 7756  df-1st 7874  df-2nd 7875  df-1o 8342  df-er 8544  df-map 8663  df-ixp 8732  df-en 8780  df-dom 8781  df-fin 8783  df-fi 9238  df-topgen 17221  df-pt 17222  df-top 22114  df-topon 22131  df-bases 22167  df-cnp 22450
This theorem is referenced by:  ptcn  22849
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