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Theorem ptpjopn 22963
Description: The projection map is an open map. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
ptpjcn.1 𝑌 = 𝐽
ptpjcn.2 𝐽 = (∏t𝐹)
Assertion
Ref Expression
ptpjopn (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) ∈ (𝐹𝐼))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐼   𝑥,𝑉   𝑥,𝑌   𝑥,𝑈
Allowed substitution hint:   𝐽(𝑥)

Proof of Theorem ptpjopn
Dummy variables 𝑔 𝑘 𝑛 𝑠 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ima 5646 . . 3 ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) = ran ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈)
2 elssuni 4898 . . . . . . 7 (𝑈𝐽𝑈 𝐽)
3 ptpjcn.1 . . . . . . 7 𝑌 = 𝐽
42, 3sseqtrrdi 3995 . . . . . 6 (𝑈𝐽𝑈𝑌)
54adantl 482 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝑈𝑌)
65resmptd 5994 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈) = (𝑥𝑈 ↦ (𝑥𝐼)))
76rneqd 5893 . . 3 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ran ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈) = ran (𝑥𝑈 ↦ (𝑥𝐼)))
81, 7eqtrid 2788 . 2 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) = ran (𝑥𝑈 ↦ (𝑥𝐼)))
9 ptpjcn.2 . . . . . . . . . . 11 𝐽 = (∏t𝐹)
10 ffn 6668 . . . . . . . . . . . 12 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
11 eqid 2736 . . . . . . . . . . . . 13 {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))} = {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))}
1211ptval 22921 . . . . . . . . . . . 12 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))}))
1310, 12sylan2 593 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))}))
149, 13eqtrid 2788 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))}))
15143adant3 1132 . . . . . . . . 9 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → 𝐽 = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))}))
1615eleq2d 2823 . . . . . . . 8 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝑈𝐽𝑈 ∈ (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))})))
1716biimpa 477 . . . . . . 7 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝑈 ∈ (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))}))
18 tg2 22315 . . . . . . 7 ((𝑈 ∈ (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))}) ∧ 𝑠𝑈) → ∃𝑤 ∈ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))} (𝑠𝑤𝑤𝑈))
1917, 18sylan 580 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → ∃𝑤 ∈ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))} (𝑠𝑤𝑤𝑈))
20 vex 3449 . . . . . . . . 9 𝑤 ∈ V
21 eqeq1 2740 . . . . . . . . . . 11 (𝑠 = 𝑤 → (𝑠 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑤 = X𝑦𝐴 (𝑔𝑦)))
2221anbi2d 629 . . . . . . . . . 10 (𝑠 = 𝑤 → (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦)) ↔ ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))))
2322exbidv 1924 . . . . . . . . 9 (𝑠 = 𝑤 → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦)) ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))))
2420, 23elab 3630 . . . . . . . 8 (𝑤 ∈ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))} ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦)))
25 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑦 = 𝐼 → (𝑔𝑦) = (𝑔𝐼))
26 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑦 = 𝐼 → (𝐹𝑦) = (𝐹𝐼))
2725, 26eleq12d 2832 . . . . . . . . . . . . . 14 (𝑦 = 𝐼 → ((𝑔𝑦) ∈ (𝐹𝑦) ↔ (𝑔𝐼) ∈ (𝐹𝐼)))
28 simplr2 1216 . . . . . . . . . . . . . 14 ((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) → ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦))
29 simpl3 1193 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝐼𝐴)
3029ad3antrrr 728 . . . . . . . . . . . . . 14 ((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) → 𝐼𝐴)
3127, 28, 30rspcdva 3582 . . . . . . . . . . . . 13 ((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) → (𝑔𝐼) ∈ (𝐹𝐼))
32 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑦 = 𝐼 → (𝑠𝑦) = (𝑠𝐼))
3332, 25eleq12d 2832 . . . . . . . . . . . . . 14 (𝑦 = 𝐼 → ((𝑠𝑦) ∈ (𝑔𝑦) ↔ (𝑠𝐼) ∈ (𝑔𝐼)))
34 vex 3449 . . . . . . . . . . . . . . . . 17 𝑠 ∈ V
3534elixp 8842 . . . . . . . . . . . . . . . 16 (𝑠X𝑦𝐴 (𝑔𝑦) ↔ (𝑠 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦)))
3635simprbi 497 . . . . . . . . . . . . . . 15 (𝑠X𝑦𝐴 (𝑔𝑦) → ∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦))
3736ad2antrl 726 . . . . . . . . . . . . . 14 ((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) → ∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦))
3833, 37, 30rspcdva 3582 . . . . . . . . . . . . 13 ((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) → (𝑠𝐼) ∈ (𝑔𝐼))
39 simplrr 776 . . . . . . . . . . . . . . . . . 18 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)
40 simplrl 775 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔𝐼) ∧ 𝑛𝐴)) ∧ 𝑛 = 𝐼) → 𝑘 ∈ (𝑔𝐼))
41 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝐼 → (𝑔𝑛) = (𝑔𝐼))
4241adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔𝐼) ∧ 𝑛𝐴)) ∧ 𝑛 = 𝐼) → (𝑔𝑛) = (𝑔𝐼))
4340, 42eleqtrrd 2841 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔𝐼) ∧ 𝑛𝐴)) ∧ 𝑛 = 𝐼) → 𝑘 ∈ (𝑔𝑛))
44 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑛 → (𝑠𝑦) = (𝑠𝑛))
45 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑛 → (𝑔𝑦) = (𝑔𝑛))
4644, 45eleq12d 2832 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑛 → ((𝑠𝑦) ∈ (𝑔𝑦) ↔ (𝑠𝑛) ∈ (𝑔𝑛)))
47 simplrl 775 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔𝐼) ∧ 𝑛𝐴)) → 𝑠X𝑦𝐴 (𝑔𝑦))
4847, 36syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔𝐼) ∧ 𝑛𝐴)) → ∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦))
49 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔𝐼) ∧ 𝑛𝐴)) → 𝑛𝐴)
5046, 48, 49rspcdva 3582 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔𝐼) ∧ 𝑛𝐴)) → (𝑠𝑛) ∈ (𝑔𝑛))
5150adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔𝐼) ∧ 𝑛𝐴)) ∧ ¬ 𝑛 = 𝐼) → (𝑠𝑛) ∈ (𝑔𝑛))
5243, 51ifclda 4521 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔𝐼) ∧ 𝑛𝐴)) → if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)) ∈ (𝑔𝑛))
5352anassrs 468 . . . . . . . . . . . . . . . . . . . . 21 ((((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) ∧ 𝑛𝐴) → if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)) ∈ (𝑔𝑛))
5453ralrimiva 3143 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → ∀𝑛𝐴 if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)) ∈ (𝑔𝑛))
55 simpll1 1212 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → 𝐴𝑉)
5655ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → 𝐴𝑉)
57 mptelixpg 8873 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑉 → ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) ∈ X𝑛𝐴 (𝑔𝑛) ↔ ∀𝑛𝐴 if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)) ∈ (𝑔𝑛)))
5856, 57syl 17 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) ∈ X𝑛𝐴 (𝑔𝑛) ↔ ∀𝑛𝐴 if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)) ∈ (𝑔𝑛)))
5954, 58mpbird 256 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) ∈ X𝑛𝐴 (𝑔𝑛))
60 fveq2 6842 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑦 → (𝑔𝑛) = (𝑔𝑦))
6160cbvixpv 8853 . . . . . . . . . . . . . . . . . . 19 X𝑛𝐴 (𝑔𝑛) = X𝑦𝐴 (𝑔𝑦)
6259, 61eleqtrdi 2848 . . . . . . . . . . . . . . . . . 18 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) ∈ X𝑦𝐴 (𝑔𝑦))
6339, 62sseldd 3945 . . . . . . . . . . . . . . . . 17 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) ∈ 𝑈)
6430adantr 481 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → 𝐼𝐴)
65 iftrue 4492 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝐼 → if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)) = 𝑘)
66 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) = (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))
67 vex 3449 . . . . . . . . . . . . . . . . . . . 20 𝑘 ∈ V
6865, 66, 67fvmpt 6948 . . . . . . . . . . . . . . . . . . 19 (𝐼𝐴 → ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼) = 𝑘)
6964, 68syl 17 . . . . . . . . . . . . . . . . . 18 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼) = 𝑘)
7069eqcomd 2742 . . . . . . . . . . . . . . . . 17 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → 𝑘 = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼))
71 fveq1 6841 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) → (𝑥𝐼) = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼))
7271rspceeqv 3595 . . . . . . . . . . . . . . . . 17 (((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) ∈ 𝑈𝑘 = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼)) → ∃𝑥𝑈 𝑘 = (𝑥𝐼))
7363, 70, 72syl2anc 584 . . . . . . . . . . . . . . . 16 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → ∃𝑥𝑈 𝑘 = (𝑥𝐼))
74 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑥𝑈 ↦ (𝑥𝐼)) = (𝑥𝑈 ↦ (𝑥𝐼))
7574elrnmpt 5911 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ V → (𝑘 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ ∃𝑥𝑈 𝑘 = (𝑥𝐼)))
7675elv 3451 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ ∃𝑥𝑈 𝑘 = (𝑥𝐼))
7773, 76sylibr 233 . . . . . . . . . . . . . . 15 (((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔𝐼)) → 𝑘 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼)))
7877ex 413 . . . . . . . . . . . . . 14 ((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) → (𝑘 ∈ (𝑔𝐼) → 𝑘 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))))
7978ssrdv 3950 . . . . . . . . . . . . 13 ((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) → (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))
80 eleq2 2826 . . . . . . . . . . . . . . 15 (𝑧 = (𝑔𝐼) → ((𝑠𝐼) ∈ 𝑧 ↔ (𝑠𝐼) ∈ (𝑔𝐼)))
81 sseq1 3969 . . . . . . . . . . . . . . 15 (𝑧 = (𝑔𝐼) → (𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
8280, 81anbi12d 631 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝐼) → (((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ((𝑠𝐼) ∈ (𝑔𝐼) ∧ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
8382rspcev 3581 . . . . . . . . . . . . 13 (((𝑔𝐼) ∈ (𝐹𝐼) ∧ ((𝑠𝐼) ∈ (𝑔𝐼) ∧ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
8431, 38, 79, 83syl12anc 835 . . . . . . . . . . . 12 ((((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) ∧ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
8584ex 413 . . . . . . . . . . 11 (((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → ((𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
86 eleq2 2826 . . . . . . . . . . . . 13 (𝑤 = X𝑦𝐴 (𝑔𝑦) → (𝑠𝑤𝑠X𝑦𝐴 (𝑔𝑦)))
87 sseq1 3969 . . . . . . . . . . . . 13 (𝑤 = X𝑦𝐴 (𝑔𝑦) → (𝑤𝑈X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈))
8886, 87anbi12d 631 . . . . . . . . . . . 12 (𝑤 = X𝑦𝐴 (𝑔𝑦) → ((𝑠𝑤𝑤𝑈) ↔ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)))
8988imbi1d 341 . . . . . . . . . . 11 (𝑤 = X𝑦𝐴 (𝑔𝑦) → (((𝑠𝑤𝑤𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))) ↔ ((𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))))
9085, 89syl5ibrcom 246 . . . . . . . . . 10 (((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → (𝑤 = X𝑦𝐴 (𝑔𝑦) → ((𝑠𝑤𝑤𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))))
9190expimpd 454 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦)) → ((𝑠𝑤𝑤𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))))
9291exlimdv 1936 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦)) → ((𝑠𝑤𝑤𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))))
9324, 92biimtrid 241 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → (𝑤 ∈ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))} → ((𝑠𝑤𝑤𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))))
9493rexlimdv 3150 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → (∃𝑤 ∈ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))} (𝑠𝑤𝑤𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
9519, 94mpd 15 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
9695ralrimiva 3143 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ∀𝑠𝑈𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
97 fvex 6855 . . . . . 6 (𝑠𝐼) ∈ V
9897rgenw 3068 . . . . 5 𝑠𝑈 (𝑠𝐼) ∈ V
99 fveq1 6841 . . . . . . 7 (𝑥 = 𝑠 → (𝑥𝐼) = (𝑠𝐼))
10099cbvmptv 5218 . . . . . 6 (𝑥𝑈 ↦ (𝑥𝐼)) = (𝑠𝑈 ↦ (𝑠𝐼))
101 eleq1 2825 . . . . . . . 8 (𝑦 = (𝑠𝐼) → (𝑦𝑧 ↔ (𝑠𝐼) ∈ 𝑧))
102101anbi1d 630 . . . . . . 7 (𝑦 = (𝑠𝐼) → ((𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
103102rexbidv 3175 . . . . . 6 (𝑦 = (𝑠𝐼) → (∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
104100, 103ralrnmptw 7044 . . . . 5 (∀𝑠𝑈 (𝑠𝐼) ∈ V → (∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ∀𝑠𝑈𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
10598, 104ax-mp 5 . . . 4 (∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ∀𝑠𝑈𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
10696, 105sylibr 233 . . 3 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
107 simpl2 1192 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝐹:𝐴⟶Top)
108107, 29ffvelcdmd 7036 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → (𝐹𝐼) ∈ Top)
109 eltop2 22325 . . . 4 ((𝐹𝐼) ∈ Top → (ran (𝑥𝑈 ↦ (𝑥𝐼)) ∈ (𝐹𝐼) ↔ ∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
110108, 109syl 17 . . 3 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → (ran (𝑥𝑈 ↦ (𝑥𝐼)) ∈ (𝐹𝐼) ↔ ∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
111106, 110mpbird 256 . 2 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ran (𝑥𝑈 ↦ (𝑥𝐼)) ∈ (𝐹𝐼))
1128, 111eqeltrd 2838 1 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) ∈ (𝐹𝐼))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  wss 3910  ifcif 4486   cuni 4865  cmpt 5188  ran crn 5634  cres 5635  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  Xcixp 8835  Fincfn 8883  topGenctg 17319  tcpt 17320  Topctop 22242
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-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-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  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-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ixp 8836  df-topgen 17325  df-pt 17326  df-top 22243
This theorem is referenced by: (None)
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