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Theorem ptpjopn 21636
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 5262 . . 3 ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) = ran ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈)
2 elssuni 4603 . . . . . . 7 (𝑈𝐽𝑈 𝐽)
3 ptpjcn.1 . . . . . . 7 𝑌 = 𝐽
42, 3syl6sseqr 3801 . . . . . 6 (𝑈𝐽𝑈𝑌)
54adantl 467 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝑈𝑌)
65resmptd 5593 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈) = (𝑥𝑈 ↦ (𝑥𝐼)))
76rneqd 5491 . . 3 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ran ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈) = ran (𝑥𝑈 ↦ (𝑥𝐼)))
81, 7syl5eq 2817 . 2 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) = ran (𝑥𝑈 ↦ (𝑥𝐼)))
9 ptpjcn.2 . . . . . . . . . . 11 𝐽 = (∏t𝐹)
10 ffn 6185 . . . . . . . . . . . 12 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
11 eqid 2771 . . . . . . . . . . . . 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 21594 . . . . . . . . . . . 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 580 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))}))
149, 13syl5eq 2817 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))}))
15143adant3 1126 . . . . . . . . 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 2836 . . . . . . . 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 462 . . . . . . 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 20990 . . . . . . 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 569 . . . . . 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 3354 . . . . . . . . 9 𝑤 ∈ V
21 eqeq1 2775 . . . . . . . . . . 11 (𝑠 = 𝑤 → (𝑠 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑤 = X𝑦𝐴 (𝑔𝑦)))
2221anbi2d 614 . . . . . . . . . 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 2002 . . . . . . . . 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 3501 . . . . . . . 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 6332 . . . . . . . . . . . . . . 15 (𝑦 = 𝐼 → (𝑔𝑦) = (𝑔𝐼))
26 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑦 = 𝐼 → (𝐹𝑦) = (𝐹𝐼))
2725, 26eleq12d 2844 . . . . . . . . . . . . . 14 (𝑦 = 𝐼 → ((𝑔𝑦) ∈ (𝐹𝑦) ↔ (𝑔𝐼) ∈ (𝐹𝐼)))
28 simplr2 1262 . . . . . . . . . . . . . 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 1231 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝐼𝐴)
3029ad3antrrr 709 . . . . . . . . . . . . . 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 3466 . . . . . . . . . . . . 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 6332 . . . . . . . . . . . . . . 15 (𝑦 = 𝐼 → (𝑠𝑦) = (𝑠𝐼))
3332, 25eleq12d 2844 . . . . . . . . . . . . . 14 (𝑦 = 𝐼 → ((𝑠𝑦) ∈ (𝑔𝑦) ↔ (𝑠𝐼) ∈ (𝑔𝐼)))
34 vex 3354 . . . . . . . . . . . . . . . . 17 𝑠 ∈ V
3534elixp 8069 . . . . . . . . . . . . . . . 16 (𝑠X𝑦𝐴 (𝑔𝑦) ↔ (𝑠 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦)))
3635simprbi 484 . . . . . . . . . . . . . . 15 (𝑠X𝑦𝐴 (𝑔𝑦) → ∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦))
3736ad2antrl 707 . . . . . . . . . . . . . 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 3466 . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . . . . . . . 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 6332 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝐼 → (𝑔𝑛) = (𝑔𝐼))
4241adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 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 2853 . . . . . . . . . . . . . . . . . . . . . . 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 6332 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑛 → (𝑠𝑦) = (𝑠𝑛))
45 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑛 → (𝑔𝑦) = (𝑔𝑛))
4644, 45eleq12d 2844 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑛 → ((𝑠𝑦) ∈ (𝑔𝑦) ↔ (𝑠𝑛) ∈ (𝑔𝑛)))
47 simplrl 762 . . . . . . . . . . . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . . . . . . . . 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 3466 . . . . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . . . 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 4259 . . . . . . . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . . . . . . . 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 3115 . . . . . . . . . . . . . . . . . . . 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 1254 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → 𝐴𝑉)
5655ad3antrrr 709 . . . . . . . . . . . . . . . . . . . . 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 8099 . . . . . . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . . . . . . 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 6332 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑦 → (𝑔𝑛) = (𝑔𝑦))
6160cbvixpv 8080 . . . . . . . . . . . . . . . . . . 19 X𝑛𝐴 (𝑔𝑛) = X𝑦𝐴 (𝑔𝑦)
6259, 61syl6eleq 2860 . . . . . . . . . . . . . . . . . 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 3753 . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . 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 4231 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝐼 → if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)) = 𝑘)
66 eqid 2771 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) = (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))
67 vex 3354 . . . . . . . . . . . . . . . . . . . 20 𝑘 ∈ V
6865, 66, 67fvmpt 6424 . . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . 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 6331 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) → (𝑥𝐼) = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼))
7271eqeq2d 2781 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) → (𝑘 = (𝑥𝐼) ↔ 𝑘 = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼)))
7372rspcev 3460 . . . . . . . . . . . . . . . . 17 (((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) ∈ 𝑈𝑘 = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼)) → ∃𝑥𝑈 𝑘 = (𝑥𝐼))
7463, 70, 73syl2anc 573 . . . . . . . . . . . . . . . 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))
75 eqid 2771 . . . . . . . . . . . . . . . . . 18 (𝑥𝑈 ↦ (𝑥𝐼)) = (𝑥𝑈 ↦ (𝑥𝐼))
7675elrnmpt 5510 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ V → (𝑘 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ ∃𝑥𝑈 𝑘 = (𝑥𝐼)))
7767, 76ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ ∃𝑥𝑈 𝑘 = (𝑥𝐼))
7874, 77sylibr 224 . . . . . . . . . . . . . . 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)))
7978ex 397 . . . . . . . . . . . . . 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))))
8079ssrdv 3758 . . . . . . . . . . . . 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)))
81 eleq2 2839 . . . . . . . . . . . . . . 15 (𝑧 = (𝑔𝐼) → ((𝑠𝐼) ∈ 𝑧 ↔ (𝑠𝐼) ∈ (𝑔𝐼)))
82 sseq1 3775 . . . . . . . . . . . . . . 15 (𝑧 = (𝑔𝐼) → (𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
8381, 82anbi12d 616 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝐼) → (((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ((𝑠𝐼) ∈ (𝑔𝐼) ∧ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
8483rspcev 3460 . . . . . . . . . . . . 13 (((𝑔𝐼) ∈ (𝐹𝐼) ∧ ((𝑠𝐼) ∈ (𝑔𝐼) ∧ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
8531, 38, 80, 84syl12anc 1474 . . . . . . . . . . . 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))))
8685ex 397 . . . . . . . . . . 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)))))
87 eleq2 2839 . . . . . . . . . . . . 13 (𝑤 = X𝑦𝐴 (𝑔𝑦) → (𝑠𝑤𝑠X𝑦𝐴 (𝑔𝑦)))
88 sseq1 3775 . . . . . . . . . . . . 13 (𝑤 = X𝑦𝐴 (𝑔𝑦) → (𝑤𝑈X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈))
8987, 88anbi12d 616 . . . . . . . . . . . 12 (𝑤 = X𝑦𝐴 (𝑔𝑦) → ((𝑠𝑤𝑤𝑈) ↔ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)))
9089imbi1d 330 . . . . . . . . . . 11 (𝑤 = X𝑦𝐴 (𝑔𝑦) → (((𝑠𝑤𝑤𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))) ↔ ((𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))))
9186, 90syl5ibrcom 237 . . . . . . . . . 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))))))
9291expimpd 441 . . . . . . . . 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))))))
9392exlimdv 2013 . . . . . . . 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))))))
9424, 93syl5bi 232 . . . . . . 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))))))
9594rexlimdv 3178 . . . . . 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)))))
9619, 95mpd 15 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
9796ralrimiva 3115 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ∀𝑠𝑈𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
98 fvex 6342 . . . . . 6 (𝑠𝐼) ∈ V
9998rgenw 3073 . . . . 5 𝑠𝑈 (𝑠𝐼) ∈ V
100 fveq1 6331 . . . . . . 7 (𝑥 = 𝑠 → (𝑥𝐼) = (𝑠𝐼))
101100cbvmptv 4884 . . . . . 6 (𝑥𝑈 ↦ (𝑥𝐼)) = (𝑠𝑈 ↦ (𝑠𝐼))
102 eleq1 2838 . . . . . . . 8 (𝑦 = (𝑠𝐼) → (𝑦𝑧 ↔ (𝑠𝐼) ∈ 𝑧))
103102anbi1d 615 . . . . . . 7 (𝑦 = (𝑠𝐼) → ((𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
104103rexbidv 3200 . . . . . 6 (𝑦 = (𝑠𝐼) → (∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
105101, 104ralrnmpt 6511 . . . . 5 (∀𝑠𝑈 (𝑠𝐼) ∈ V → (∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ∀𝑠𝑈𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
10699, 105ax-mp 5 . . . 4 (∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ∀𝑠𝑈𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
10797, 106sylibr 224 . . 3 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
108 simpl2 1229 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝐹:𝐴⟶Top)
109108, 29ffvelrnd 6503 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → (𝐹𝐼) ∈ Top)
110 eltop2 21000 . . . 4 ((𝐹𝐼) ∈ Top → (ran (𝑥𝑈 ↦ (𝑥𝐼)) ∈ (𝐹𝐼) ↔ ∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
111109, 110syl 17 . . 3 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → (ran (𝑥𝑈 ↦ (𝑥𝐼)) ∈ (𝐹𝐼) ↔ ∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
112107, 111mpbird 247 . 2 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ran (𝑥𝑈 ↦ (𝑥𝐼)) ∈ (𝐹𝐼))
1138, 112eqeltrd 2850 1 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) ∈ (𝐹𝐼))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  wss 3723  ifcif 4225   cuni 4574  cmpt 4863  ran crn 5250  cres 5251  cima 5252   Fn wfn 6026  wf 6027  cfv 6031  Xcixp 8062  Fincfn 8109  topGenctg 16306  tcpt 16307  Topctop 20918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ixp 8063  df-topgen 16312  df-pt 16313  df-top 20919
This theorem is referenced by: (None)
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