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Theorem ptpjopn 22873
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 5640 . . 3 ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) = ran ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈)
2 elssuni 4893 . . . . . . 7 (𝑈𝐽𝑈 𝐽)
3 ptpjcn.1 . . . . . . 7 𝑌 = 𝐽
42, 3sseqtrrdi 3990 . . . . . 6 (𝑈𝐽𝑈𝑌)
54adantl 483 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝑈𝑌)
65resmptd 5987 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈) = (𝑥𝑈 ↦ (𝑥𝐼)))
76rneqd 5886 . . 3 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ran ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈) = ran (𝑥𝑈 ↦ (𝑥𝐼)))
81, 7eqtrid 2789 . 2 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) = ran (𝑥𝑈 ↦ (𝑥𝐼)))
9 ptpjcn.2 . . . . . . . . . . 11 𝐽 = (∏t𝐹)
10 ffn 6660 . . . . . . . . . . . 12 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
11 eqid 2737 . . . . . . . . . . . . 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 22831 . . . . . . . . . . . 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 594 . . . . . . . . . . 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 2789 . . . . . . . . . 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 478 . . . . . . 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 22225 . . . . . . 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 581 . . . . . 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 3447 . . . . . . . . 9 𝑤 ∈ V
21 eqeq1 2741 . . . . . . . . . . 11 (𝑠 = 𝑤 → (𝑠 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑤 = X𝑦𝐴 (𝑔𝑦)))
2221anbi2d 630 . . . . . . . . . 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 3625 . . . . . . . 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 6834 . . . . . . . . . . . . . . 15 (𝑦 = 𝐼 → (𝑔𝑦) = (𝑔𝐼))
26 fveq2 6834 . . . . . . . . . . . . . . 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 3577 . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . 15 (𝑦 = 𝐼 → (𝑠𝑦) = (𝑠𝐼))
3332, 25eleq12d 2832 . . . . . . . . . . . . . 14 (𝑦 = 𝐼 → ((𝑠𝑦) ∈ (𝑔𝑦) ↔ (𝑠𝐼) ∈ (𝑔𝐼)))
34 vex 3447 . . . . . . . . . . . . . . . . 17 𝑠 ∈ V
3534elixp 8772 . . . . . . . . . . . . . . . 16 (𝑠X𝑦𝐴 (𝑔𝑦) ↔ (𝑠 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦)))
3635simprbi 498 . . . . . . . . . . . . . . 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 3577 . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝐼 → (𝑔𝑛) = (𝑔𝐼))
4241adantl 483 . . . . . . . . . . . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑛 → (𝑠𝑦) = (𝑠𝑛))
45 fveq2 6834 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3577 . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . 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 4516 . . . . . . . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . . . . . . 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 3141 . . . . . . . . . . . . . . . . . . . 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 8803 . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑦 → (𝑔𝑛) = (𝑔𝑦))
6160cbvixpv 8783 . . . . . . . . . . . . . . . . . . 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 3940 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . 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 4487 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝐼 → if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)) = 𝑘)
66 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) = (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))
67 vex 3447 . . . . . . . . . . . . . . . . . . . 20 𝑘 ∈ V
6865, 66, 67fvmpt 6940 . . . . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . . . 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 6833 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) → (𝑥𝐼) = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼))
7271rspceeqv 3590 . . . . . . . . . . . . . . . . 17 (((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) ∈ 𝑈𝑘 = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼)) → ∃𝑥𝑈 𝑘 = (𝑥𝐼))
7363, 70, 72syl2anc 585 . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . 18 (𝑥𝑈 ↦ (𝑥𝐼)) = (𝑥𝑈 ↦ (𝑥𝐼))
7574elrnmpt 5904 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ V → (𝑘 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ ∃𝑥𝑈 𝑘 = (𝑥𝐼)))
7675elv 3449 . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . 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 3945 . . . . . . . . . . . . 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 3964 . . . . . . . . . . . . . . 15 (𝑧 = (𝑔𝐼) → (𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
8280, 81anbi12d 632 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝐼) → (((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ((𝑠𝐼) ∈ (𝑔𝐼) ∧ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
8382rspcev 3576 . . . . . . . . . . . . 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 414 . . . . . . . . . . 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 3964 . . . . . . . . . . . . 13 (𝑤 = X𝑦𝐴 (𝑔𝑦) → (𝑤𝑈X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈))
8886, 87anbi12d 632 . . . . . . . . . . . 12 (𝑤 = X𝑦𝐴 (𝑔𝑦) → ((𝑠𝑤𝑤𝑈) ↔ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)))
8988imbi1d 342 . . . . . . . . . . 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 247 . . . . . . . . . 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 455 . . . . . . . . 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 3148 . . . . . 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 3141 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ∀𝑠𝑈𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
97 fvex 6847 . . . . . 6 (𝑠𝐼) ∈ V
9897rgenw 3066 . . . . 5 𝑠𝑈 (𝑠𝐼) ∈ V
99 fveq1 6833 . . . . . . 7 (𝑥 = 𝑠 → (𝑥𝐼) = (𝑠𝐼))
10099cbvmptv 5213 . . . . . 6 (𝑥𝑈 ↦ (𝑥𝐼)) = (𝑠𝑈 ↦ (𝑠𝐼))
101 eleq1 2825 . . . . . . . 8 (𝑦 = (𝑠𝐼) → (𝑦𝑧 ↔ (𝑠𝐼) ∈ 𝑧))
102101anbi1d 631 . . . . . . 7 (𝑦 = (𝑠𝐼) → ((𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
103102rexbidv 3173 . . . . . 6 (𝑦 = (𝑠𝐼) → (∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
104100, 103ralrnmptw 7035 . . . . 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 7027 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → (𝐹𝐼) ∈ Top)
109 eltop2 22235 . . . 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 257 . 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 397  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2714  wral 3062  wrex 3071  Vcvv 3443  cdif 3902  wss 3905  ifcif 4481   cuni 4860  cmpt 5183  ran crn 5628  cres 5629  cima 5630   Fn wfn 6483  wf 6484  cfv 6488  Xcixp 8765  Fincfn 8813  topGenctg 17250  tcpt 17251  Topctop 22152
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 2708  ax-rep 5237  ax-sep 5251  ax-nul 5258  ax-pow 5315  ax-pr 5379  ax-un 7659
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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 3352  df-rab 3406  df-v 3445  df-sbc 3735  df-csb 3851  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4278  df-if 4482  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5184  df-id 5525  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ixp 8766  df-topgen 17256  df-pt 17257  df-top 22153
This theorem is referenced by: (None)
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