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Theorem ptcnp 23631
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 480 . . . . . . . 8 ((𝜑𝑘𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3 ptcnp.5 . . . . . . . . . 10 (𝜑𝐹:𝐼⟶Top)
43ffvelcdmda 7103 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
5 toptopon2 22925 . . . . . . . . 9 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
64, 5sylib 218 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
7 ptcnp.7 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))
8 cnpf2 23259 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷)) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
92, 6, 7, 8syl3anc 1372 . . . . . . 7 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
109fvmptelcdm 7132 . . . . . 6 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
1110an32s 652 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → 𝐴 (𝐹𝑘))
1211ralrimiva 3145 . . . 4 ((𝜑𝑥𝑋) → ∀𝑘𝐼 𝐴 (𝐹𝑘))
13 ptcnp.4 . . . . . 6 (𝜑𝐼𝑉)
1413adantr 480 . . . . 5 ((𝜑𝑥𝑋) → 𝐼𝑉)
15 mptelixpg 8976 . . . . 5 (𝐼𝑉 → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
1614, 15syl 17 . . . 4 ((𝜑𝑥𝑋) → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
1712, 16mpbird 257 . . 3 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘))
1817fmpttd 7134 . 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 1913 . . . . . . . . . . . . . 14 𝑘(𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))
23 nfv 1913 . . . . . . . . . . . . . . 15 𝑘(𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))
24 nfcv 2904 . . . . . . . . . . . . . . . . . 18 𝑘𝑋
25 nfmpt1 5249 . . . . . . . . . . . . . . . . . 18 𝑘(𝑘𝐼𝐴)
2624, 25nfmpt 5248 . . . . . . . . . . . . . . . . 17 𝑘(𝑥𝑋 ↦ (𝑘𝐼𝐴))
27 nfcv 2904 . . . . . . . . . . . . . . . . 17 𝑘𝐷
2826, 27nffv 6915 . . . . . . . . . . . . . . . 16 𝑘((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
2928nfel1 2921 . . . . . . . . . . . . . . 15 𝑘((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)
3023, 29nfan 1898 . . . . . . . . . . . . . 14 𝑘((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛))
3122, 30nfan 1898 . . . . . . . . . . . . 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 6905 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑔𝑛) = (𝑔𝑘))
35 fveq2 6905 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
3634, 35eleq12d 2834 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑔𝑛) ∈ (𝐹𝑛) ↔ (𝑔𝑘) ∈ (𝐹𝑘)))
3736rspccva 3620 . . . . . . . . . . . . . 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 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 494 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → 𝑤 ∈ Fin)
4139simprd 495 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))
4235unieqd 4919 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
4334, 42eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑔𝑛) = (𝐹𝑛) ↔ (𝑔𝑘) = (𝐹𝑘)))
4443rspccva 3620 . . . . . . . . . . . . . 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 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 8956 . . . . . . . . . . . . . 14 X𝑛𝐼 (𝑔𝑛) = X𝑘𝐼 (𝑔𝑘)
4846, 47eleqtrdi 2850 . . . . . . . . . . . . 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 23630 . . . . . . . . . . . 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 467 . . . . . . . . . . 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 456 . . . . . . . . . 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 3155 . . . . . . . . 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 454 . . . . . . . 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 2829 . . . . . . . 8 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))
5647eqeq2i 2749 . . . . . . . . . . . 12 (𝑓 = X𝑛𝐼 (𝑔𝑛) ↔ 𝑓 = X𝑘𝐼 (𝑔𝑘))
5756biimpi 216 . . . . . . . . . . 11 (𝑓 = X𝑛𝐼 (𝑔𝑛) → 𝑓 = X𝑘𝐼 (𝑔𝑘))
5857sseq2d 4015 . . . . . . . . . 10 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓 ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘)))
5958anbi2d 630 . . . . . . . . 9 (𝑓 = X𝑛𝐼 (𝑔𝑛) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓) ↔ (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘))))
6059rexbidv 3178 . . . . . . . 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 247 . . . . . 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 453 . . . . 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 1932 . . . 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 1926 . . 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 630 . . . . 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 1920 . . . 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 3696 . . 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 234 . 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 6736 . . . . 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 23579 . . . . 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 6976 . . . . . 6 (𝜑𝐹 = (𝑘𝐼 ↦ (𝐹𝑘)))
7776fveq2d 6909 . . . . 5 (𝜑 → (∏t𝐹) = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))))
7820, 77eqtrid 2788 . . . 4 (𝜑𝐾 = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))))
796ralrimiva 3145 . . . . 5 (𝜑 → ∀𝑘𝐼 (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
80 eqid 2736 . . . . . 6 (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘)))
8180pttopon 23605 . . . . 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 2840 . . 3 (𝜑𝐾 ∈ (TopOn‘X𝑘𝐼 (𝐹𝑘)))
841, 75, 83, 21tgcnp 23262 . 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 713 1 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1537   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wral 3060  wrex 3069  cdif 3947  wss 3950   cuni 4906  cmpt 5224  cima 5687   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  Xcixp 8938  Fincfn 8986  topGenctg 17483  tcpt 17484  Topctop 22900  TopOnctopon 22917   CnP ccnp 23234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-1o 8507  df-2o 8508  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-fin 8990  df-fi 9452  df-topgen 17489  df-pt 17490  df-top 22901  df-topon 22918  df-bases 22954  df-cnp 23237
This theorem is referenced by:  ptcn  23636
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