MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ptcnp Structured version   Visualization version   GIF version

Theorem ptcnp 21627
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 . . . . . . . . . 10 (𝜑𝐽 ∈ (TopOn‘𝑋))
21adantr 472 . . . . . . . . 9 ((𝜑𝑘𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3 ptcnp.5 . . . . . . . . . . 11 (𝜑𝐹:𝐼⟶Top)
43ffvelrnda 6522 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
5 eqid 2760 . . . . . . . . . . 11 (𝐹𝑘) = (𝐹𝑘)
65toptopon 20924 . . . . . . . . . 10 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
74, 6sylib 208 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
8 ptcnp.7 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))
9 cnpf2 21256 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷)) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
102, 7, 8, 9syl3anc 1477 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
11 eqid 2760 . . . . . . . . 9 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1211fmpt 6544 . . . . . . . 8 (∀𝑥𝑋 𝐴 (𝐹𝑘) ↔ (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
1310, 12sylibr 224 . . . . . . 7 ((𝜑𝑘𝐼) → ∀𝑥𝑋 𝐴 (𝐹𝑘))
1413r19.21bi 3070 . . . . . 6 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
1514an32s 881 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → 𝐴 (𝐹𝑘))
1615ralrimiva 3104 . . . 4 ((𝜑𝑥𝑋) → ∀𝑘𝐼 𝐴 (𝐹𝑘))
17 ptcnp.4 . . . . . 6 (𝜑𝐼𝑉)
1817adantr 472 . . . . 5 ((𝜑𝑥𝑋) → 𝐼𝑉)
19 mptelixpg 8111 . . . . 5 (𝐼𝑉 → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
2018, 19syl 17 . . . 4 ((𝜑𝑥𝑋) → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
2116, 20mpbird 247 . . 3 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘))
22 eqid 2760 . . 3 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) = (𝑥𝑋 ↦ (𝑘𝐼𝐴))
2321, 22fmptd 6548 . 2 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋X𝑘𝐼 (𝐹𝑘))
24 df-3an 1074 . . . . . . . 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)))
25 ptcnp.2 . . . . . . . . . . . . 13 𝐾 = (∏t𝐹)
26 ptcnp.6 . . . . . . . . . . . . 13 (𝜑𝐷𝑋)
27 nfv 1992 . . . . . . . . . . . . . 14 𝑘(𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))
28 nfv 1992 . . . . . . . . . . . . . . 15 𝑘(𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))
29 nfcv 2902 . . . . . . . . . . . . . . . . . 18 𝑘𝑋
30 nfmpt1 4899 . . . . . . . . . . . . . . . . . 18 𝑘(𝑘𝐼𝐴)
3129, 30nfmpt 4898 . . . . . . . . . . . . . . . . 17 𝑘(𝑥𝑋 ↦ (𝑘𝐼𝐴))
32 nfcv 2902 . . . . . . . . . . . . . . . . 17 𝑘𝐷
3331, 32nffv 6359 . . . . . . . . . . . . . . . 16 𝑘((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
3433nfel1 2917 . . . . . . . . . . . . . . 15 𝑘((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)
3528, 34nfan 1977 . . . . . . . . . . . . . 14 𝑘((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛))
3627, 35nfan 1977 . . . . . . . . . . . . 13 𝑘((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))
37 simprll 821 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → 𝑔 Fn 𝐼)
38 simprlr 822 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛))
39 fveq2 6352 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑔𝑛) = (𝑔𝑘))
40 fveq2 6352 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
4139, 40eleq12d 2833 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑔𝑛) ∈ (𝐹𝑛) ↔ (𝑔𝑘) ∈ (𝐹𝑘)))
4241rspccva 3448 . . . . . . . . . . . . . 14 ((∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ 𝑘𝐼) → (𝑔𝑘) ∈ (𝐹𝑘))
4338, 42sylan 489 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) ∧ 𝑘𝐼) → (𝑔𝑘) ∈ (𝐹𝑘))
44 simprrl 823 . . . . . . . . . . . . . 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)))
4544simpld 477 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → 𝑤 ∈ Fin)
4644simprd 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛))
4740unieqd 4598 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
4839, 47eqeq12d 2775 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑔𝑛) = (𝐹𝑛) ↔ (𝑔𝑘) = (𝐹𝑘)))
4948rspccva 3448 . . . . . . . . . . . . . 14 ((∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛) ∧ 𝑘 ∈ (𝐼𝑤)) → (𝑔𝑘) = (𝐹𝑘))
5046, 49sylan 489 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) ∧ 𝑘 ∈ (𝐼𝑤)) → (𝑔𝑘) = (𝐹𝑘))
51 simprrr 824 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛))
5239cbvixpv 8092 . . . . . . . . . . . . . 14 X𝑛𝐼 (𝑔𝑛) = X𝑘𝐼 (𝑔𝑘)
5351, 52syl6eleq 2849 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑘𝐼 (𝑔𝑘))
5425, 1, 17, 3, 26, 8, 36, 37, 43, 45, 50, 53ptcnplem 21626 . . . . . . . . . . . 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)))
5554anassrs 683 . . . . . . . . . . 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)))
5655expr 644 . . . . . . . . . 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))))
5756rexlimdvaa 3170 . . . . . . . . 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)))))
5857impr 650 . . . . . . . 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))))
5924, 58sylan2b 493 . . . . . . 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))))
60 eleq2 2828 . . . . . . . 8 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ 𝑓 ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑛𝐼 (𝑔𝑛)))
6152eqeq2i 2772 . . . . . . . . . . . 12 (𝑓 = X𝑛𝐼 (𝑔𝑛) ↔ 𝑓 = X𝑘𝐼 (𝑔𝑘))
6261biimpi 206 . . . . . . . . . . 11 (𝑓 = X𝑛𝐼 (𝑔𝑛) → 𝑓 = X𝑘𝐼 (𝑔𝑘))
6362sseq2d 3774 . . . . . . . . . 10 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓 ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘)))
6463anbi2d 742 . . . . . . . . 9 (𝑓 = X𝑛𝐼 (𝑔𝑛) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓) ↔ (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘))))
6564rexbidv 3190 . . . . . . . 8 (𝑓 = X𝑛𝐼 (𝑔𝑛) → (∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ 𝑓) ↔ ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝑔𝑘))))
6660, 65imbi12d 333 . . . . . . 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)))))
6759, 66syl5ibrcom 237 . . . . . 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))))
6867expimpd 630 . . . . 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))))
6968exlimdv 2010 . . . 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))))
7069alrimiv 2004 . . 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))))
71 eqeq1 2764 . . . . . 6 (𝑎 = 𝑓 → (𝑎 = X𝑛𝐼 (𝑔𝑛) ↔ 𝑓 = X𝑛𝐼 (𝑔𝑛)))
7271anbi2d 742 . . . . 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))))
7372exbidv 1999 . . . 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))))
7473ralab 3508 . . 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))))
7570, 74sylibr 224 . 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)))
76 ffn 6206 . . . . . 6 (𝐹:𝐼⟶Top → 𝐹 Fn 𝐼)
773, 76syl 17 . . . . 5 (𝜑𝐹 Fn 𝐼)
78 eqid 2760 . . . . . 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))}
7978ptval 21575 . . . . 5 ((𝐼𝑉𝐹 Fn 𝐼) → (∏t𝐹) = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))}))
8017, 77, 79syl2anc 696 . . . 4 (𝜑 → (∏t𝐹) = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))}))
8125, 80syl5eq 2806 . . 3 (𝜑𝐾 = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛𝐼 (𝑔𝑛) ∈ (𝐹𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼𝑤)(𝑔𝑛) = (𝐹𝑛)) ∧ 𝑎 = X𝑛𝐼 (𝑔𝑛))}))
823feqmptd 6411 . . . . . 6 (𝜑𝐹 = (𝑘𝐼 ↦ (𝐹𝑘)))
8382fveq2d 6356 . . . . 5 (𝜑 → (∏t𝐹) = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))))
8425, 83syl5eq 2806 . . . 4 (𝜑𝐾 = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))))
857ralrimiva 3104 . . . . 5 (𝜑 → ∀𝑘𝐼 (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
86 eqid 2760 . . . . . 6 (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) = (∏t‘(𝑘𝐼 ↦ (𝐹𝑘)))
8786pttopon 21601 . . . . 5 ((𝐼𝑉 ∧ ∀𝑘𝐼 (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘))) → (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) ∈ (TopOn‘X𝑘𝐼 (𝐹𝑘)))
8817, 85, 87syl2anc 696 . . . 4 (𝜑 → (∏t‘(𝑘𝐼 ↦ (𝐹𝑘))) ∈ (TopOn‘X𝑘𝐼 (𝐹𝑘)))
8984, 88eqeltrd 2839 . . 3 (𝜑𝐾 ∈ (TopOn‘X𝑘𝐼 (𝐹𝑘)))
901, 81, 89, 26tgcnp 21259 . 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)))))
9123, 75, 90mpbir2and 995 1 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wal 1630   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wral 3050  wrex 3051  cdif 3712  wss 3715   cuni 4588  cmpt 4881  cima 5269   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  Xcixp 8074  Fincfn 8121  topGenctg 16300  tcpt 16301  Topctop 20900  TopOnctopon 20917   CnP ccnp 21231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-ixp 8075  df-en 8122  df-dom 8123  df-fin 8125  df-fi 8482  df-topgen 16306  df-pt 16307  df-top 20901  df-topon 20918  df-bases 20952  df-cnp 21234
This theorem is referenced by:  ptcn  21632
  Copyright terms: Public domain W3C validator