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Theorem ptcnp 22973
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 7035 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
5 toptopon2 22267 . . . . . . . . 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 22601 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷)) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
92, 6, 7, 8syl3anc 1371 . . . . . . 7 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
109fvmptelcdm 7061 . . . . . 6 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
1110an32s 650 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → 𝐴 (𝐹𝑘))
1211ralrimiva 3143 . . . 4 ((𝜑𝑥𝑋) → ∀𝑘𝐼 𝐴 (𝐹𝑘))
13 ptcnp.4 . . . . . 6 (𝜑𝐼𝑉)
1413adantr 481 . . . . 5 ((𝜑𝑥𝑋) → 𝐼𝑉)
15 mptelixpg 8873 . . . . 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 7063 . 2 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋X𝑘𝐼 (𝐹𝑘))
19 df-3an 1089 . . . . . . . 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 1917 . . . . . . . . . . . . . 14 𝑘(𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))
23 nfv 1917 . . . . . . . . . . . . . . 15 𝑘(𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))
24 nfcv 2907 . . . . . . . . . . . . . . . . . 18 𝑘𝑋
25 nfmpt1 5213 . . . . . . . . . . . . . . . . . 18 𝑘(𝑘𝐼𝐴)
2624, 25nfmpt 5212 . . . . . . . . . . . . . . . . 17 𝑘(𝑥𝑋 ↦ (𝑘𝐼𝐴))
27 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑘𝐷
2826, 27nffv 6852 . . . . . . . . . . . . . . . 16 𝑘((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
2928nfel1 2923 . . . . . . . . . . . . . . 15 𝑘((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)
3023, 29nfan 1902 . . . . . . . . . . . . . 14 𝑘((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛))
3122, 30nfan 1902 . . . . . . . . . . . . 13 𝑘((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))
32 simprll 777 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → 𝑔 Fn 𝐼)
33 simprlr 778 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))
34 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑔𝑛) = (𝑔𝑘))
35 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
3634, 35eleq12d 2832 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑔𝑛) ∈ (𝐹𝑛) ↔ (𝑔𝑘) ∈ (𝐹𝑘)))
3736rspccva 3580 . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . 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 4879 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
4334, 42eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑔𝑛) = (𝐹𝑛) ↔ (𝑔𝑘) = (𝐹𝑘)))
4443rspccva 3580 . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . 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 8853 . . . . . . . . . . . . . 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 22972 . . . . . . . . . . . 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 3153 . . . . . . . . 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 2749 . . . . . . . . . . . 12 (𝑓 = X𝑛𝐼 (𝑔𝑛) ↔ 𝑓 = X𝑘𝐼 (𝑔𝑘))
5756biimpi 215 . . . . . . . . . . 11 (𝑓 = X𝑛𝐼 (𝑔𝑛) → 𝑓 = X𝑘𝐼 (𝑔𝑘))
5857sseq2d 3976 . . . . . . . . . 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 3175 . . . . . . . 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 1936 . . . 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 1930 . . 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 2740 . . . . . 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 1924 . . . 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 3649 . . 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 6669 . . . . 5 (𝜑𝐹 Fn 𝐼)
72 eqid 2736 . . . . . 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 22921 . . . . 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 2788 . . 3 (𝜑𝐾 = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))}))
763feqmptd 6910 . . . . . 6 (𝜑𝐹 = (𝑘𝐼 ↦ (𝐹𝑘)))
7776fveq2d 6846 . . . . 5 (𝜑 → (∏t𝐹) = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))))
7820, 77eqtrid 2788 . . . 4 (𝜑𝐾 = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))))
796ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑘𝐼 (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
80 eqid 2736 . . . . . 6 (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘)))
8180pttopon 22947 . . . . 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 22604 . 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 711 1 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  wrex 3073  cdif 3907  wss 3910   cuni 4865  cmpt 5188  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Xcixp 8835  Fincfn 8883  topGenctg 17319  tcpt 17320  Topctop 22242  TopOnctopon 22259   CnP ccnp 22576
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-topgen 17325  df-pt 17326  df-top 22243  df-topon 22260  df-bases 22296  df-cnp 22579
This theorem is referenced by:  ptcn  22978
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