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Theorem ptpjopn 21325
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 5087 . . 3 ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) = ran ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈)
2 elssuni 4433 . . . . . . 7 (𝑈𝐽𝑈 𝐽)
3 ptpjcn.1 . . . . . . 7 𝑌 = 𝐽
42, 3syl6sseqr 3631 . . . . . 6 (𝑈𝐽𝑈𝑌)
54adantl 482 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝑈𝑌)
65resmptd 5411 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈) = (𝑥𝑈 ↦ (𝑥𝐼)))
76rneqd 5313 . . 3 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ran ((𝑥𝑌 ↦ (𝑥𝐼)) ↾ 𝑈) = ran (𝑥𝑈 ↦ (𝑥𝐼)))
81, 7syl5eq 2667 . 2 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) = ran (𝑥𝑈 ↦ (𝑥𝐼)))
9 ptpjcn.2 . . . . . . . . . . 11 𝐽 = (∏t𝐹)
10 ffn 6002 . . . . . . . . . . . 12 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
11 eqid 2621 . . . . . . . . . . . . 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 21283 . . . . . . . . . . . 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 491 . . . . . . . . . . 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 2667 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑔𝑦))}))
15143adant3 1079 . . . . . . . . 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 2684 . . . . . . . 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 501 . . . . . . 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 20680 . . . . . . 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 488 . . . . . 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 3189 . . . . . . . . 9 𝑤 ∈ V
21 eqeq1 2625 . . . . . . . . . . 11 (𝑠 = 𝑤 → (𝑠 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑤 = X𝑦𝐴 (𝑔𝑦)))
2221anbi2d 739 . . . . . . . . . 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 1847 . . . . . . . . 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 3333 . . . . . . . 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 simpl3 1064 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝐼𝐴)
2625ad3antrrr 765 . . . . . . . . . . . . . 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)
27 simplr2 1102 . . . . . . . . . . . . . 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))
28 fveq2 6148 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → (𝑔𝑦) = (𝑔𝐼))
29 fveq2 6148 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → (𝐹𝑦) = (𝐹𝐼))
3028, 29eleq12d 2692 . . . . . . . . . . . . . . 15 (𝑦 = 𝐼 → ((𝑔𝑦) ∈ (𝐹𝑦) ↔ (𝑔𝐼) ∈ (𝐹𝐼)))
3130rspcv 3291 . . . . . . . . . . . . . 14 (𝐼𝐴 → (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) → (𝑔𝐼) ∈ (𝐹𝐼)))
3226, 27, 31sylc 65 . . . . . . . . . . . . 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))
33 vex 3189 . . . . . . . . . . . . . . . . 17 𝑠 ∈ V
3433elixp 7859 . . . . . . . . . . . . . . . 16 (𝑠X𝑦𝐴 (𝑔𝑦) ↔ (𝑠 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦)))
3534simprbi 480 . . . . . . . . . . . . . . 15 (𝑠X𝑦𝐴 (𝑔𝑦) → ∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦))
3635ad2antrl 763 . . . . . . . . . . . . . 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))
37 fveq2 6148 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → (𝑠𝑦) = (𝑠𝐼))
3837, 28eleq12d 2692 . . . . . . . . . . . . . . 15 (𝑦 = 𝐼 → ((𝑠𝑦) ∈ (𝑔𝑦) ↔ (𝑠𝐼) ∈ (𝑔𝐼)))
3938rspcv 3291 . . . . . . . . . . . . . 14 (𝐼𝐴 → (∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦) → (𝑠𝐼) ∈ (𝑔𝐼)))
4026, 36, 39sylc 65 . . . . . . . . . . . . 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))
41 simplrr 800 . . . . . . . . . . . . . . . . . 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)
42 simplrl 799 . . . . . . . . . . . . . . . . . . . . . . . 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))
43 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝐼 → (𝑔𝑛) = (𝑔𝐼))
4443adantl 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))
4542, 44eleqtrrd 2701 . . . . . . . . . . . . . . . . . . . . . . 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))
46 simprr 795 . . . . . . . . . . . . . . . . . . . . . . . . 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)
47 simplrl 799 . . . . . . . . . . . . . . . . . . . . . . . . . 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, 35syl 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 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑛 → (𝑠𝑦) = (𝑠𝑛))
50 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑛 → (𝑔𝑦) = (𝑔𝑛))
5149, 50eleq12d 2692 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑛 → ((𝑠𝑦) ∈ (𝑔𝑦) ↔ (𝑠𝑛) ∈ (𝑔𝑛)))
5251rspcv 3291 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛𝐴 → (∀𝑦𝐴 (𝑠𝑦) ∈ (𝑔𝑦) → (𝑠𝑛) ∈ (𝑔𝑛)))
5346, 48, 52sylc 65 . . . . . . . . . . . . . . . . . . . . . . . 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))
5453adantr 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))
5545, 54ifclda 4092 . . . . . . . . . . . . . . . . . . . . . 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))
5655anassrs 679 . . . . . . . . . . . . . . . . . . . . 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))
5756ralrimiva 2960 . . . . . . . . . . . . . . . . . . . 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))
58 simpll1 1098 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → 𝐴𝑉)
5958ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . 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)
60 mptelixpg 7889 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑉 → ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) ∈ X𝑛𝐴 (𝑔𝑛) ↔ ∀𝑛𝐴 if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)) ∈ (𝑔𝑛)))
6159, 60syl 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)))
6257, 61mpbird 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))
63 fveq2 6148 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑦 → (𝑔𝑛) = (𝑔𝑦))
6463cbvixpv 7870 . . . . . . . . . . . . . . . . . . 19 X𝑛𝐴 (𝑔𝑛) = X𝑦𝐴 (𝑔𝑦)
6562, 64syl6eleq 2708 . . . . . . . . . . . . . . . . . 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))
6641, 65sseldd 3584 . . . . . . . . . . . . . . . . 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)
6726adantr 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)
68 iftrue 4064 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝐼 → if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)) = 𝑘)
69 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) = (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))
70 vex 3189 . . . . . . . . . . . . . . . . . . . 20 𝑘 ∈ V
7168, 69, 70fvmpt 6239 . . . . . . . . . . . . . . . . . . 19 (𝐼𝐴 → ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼) = 𝑘)
7267, 71syl 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)
7372eqcomd 2627 . . . . . . . . . . . . . . . . 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))
74 fveq1 6147 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) → (𝑥𝐼) = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼))
7574eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) → (𝑘 = (𝑥𝐼) ↔ 𝑘 = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼)))
7675rspcev 3295 . . . . . . . . . . . . . . . . 17 (((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛))) ∈ 𝑈𝑘 = ((𝑛𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠𝑛)))‘𝐼)) → ∃𝑥𝑈 𝑘 = (𝑥𝐼))
7766, 73, 76syl2anc 692 . . . . . . . . . . . . . . . 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))
78 eqid 2621 . . . . . . . . . . . . . . . . . 18 (𝑥𝑈 ↦ (𝑥𝐼)) = (𝑥𝑈 ↦ (𝑥𝐼))
7978elrnmpt 5332 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ V → (𝑘 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ ∃𝑥𝑈 𝑘 = (𝑥𝐼)))
8070, 79ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ ∃𝑥𝑈 𝑘 = (𝑥𝐼))
8177, 80sylibr 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)))
8281ex 450 . . . . . . . . . . . . . 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))))
8382ssrdv 3589 . . . . . . . . . . . . 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)))
84 eleq2 2687 . . . . . . . . . . . . . . 15 (𝑧 = (𝑔𝐼) → ((𝑠𝐼) ∈ 𝑧 ↔ (𝑠𝐼) ∈ (𝑔𝐼)))
85 sseq1 3605 . . . . . . . . . . . . . . 15 (𝑧 = (𝑔𝐼) → (𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)) ↔ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
8684, 85anbi12d 746 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝐼) → (((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ((𝑠𝐼) ∈ (𝑔𝐼) ∧ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
8786rspcev 3295 . . . . . . . . . . . . 13 (((𝑔𝐼) ∈ (𝐹𝐼) ∧ ((𝑠𝐼) ∈ (𝑔𝐼) ∧ (𝑔𝐼) ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
8832, 40, 83, 87syl12anc 1321 . . . . . . . . . . . 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))))
8988ex 450 . . . . . . . . . . 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)))))
90 eleq2 2687 . . . . . . . . . . . . 13 (𝑤 = X𝑦𝐴 (𝑔𝑦) → (𝑠𝑤𝑠X𝑦𝐴 (𝑔𝑦)))
91 sseq1 3605 . . . . . . . . . . . . 13 (𝑤 = X𝑦𝐴 (𝑔𝑦) → (𝑤𝑈X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈))
9290, 91anbi12d 746 . . . . . . . . . . . 12 (𝑤 = X𝑦𝐴 (𝑔𝑦) → ((𝑠𝑤𝑤𝑈) ↔ (𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈)))
9392imbi1d 331 . . . . . . . . . . 11 (𝑤 = X𝑦𝐴 (𝑔𝑦) → (((𝑠𝑤𝑤𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))) ↔ ((𝑠X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ 𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))))
9489, 93syl5ibrcom 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))))))
9594expimpd 628 . . . . . . . . 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))))))
9695exlimdv 1858 . . . . . . . 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))))))
9724, 96syl5bi 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))))))
9897rexlimdv 3023 . . . . . 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)))))
9919, 98mpd 15 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) ∧ 𝑠𝑈) → ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
10099ralrimiva 2960 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ∀𝑠𝑈𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
101 fvex 6158 . . . . . 6 (𝑠𝐼) ∈ V
102101rgenw 2919 . . . . 5 𝑠𝑈 (𝑠𝐼) ∈ V
103 fveq1 6147 . . . . . . 7 (𝑥 = 𝑠 → (𝑥𝐼) = (𝑠𝐼))
104103cbvmptv 4710 . . . . . 6 (𝑥𝑈 ↦ (𝑥𝐼)) = (𝑠𝑈 ↦ (𝑠𝐼))
105 eleq1 2686 . . . . . . . 8 (𝑦 = (𝑠𝐼) → (𝑦𝑧 ↔ (𝑠𝐼) ∈ 𝑧))
106105anbi1d 740 . . . . . . 7 (𝑦 = (𝑠𝐼) → ((𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
107106rexbidv 3045 . . . . . 6 (𝑦 = (𝑠𝐼) → (∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ∃𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
108104, 107ralrnmpt 6324 . . . . 5 (∀𝑠𝑈 (𝑠𝐼) ∈ V → (∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ∀𝑠𝑈𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
109102, 108ax-mp 5 . . . 4 (∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))) ↔ ∀𝑠𝑈𝑧 ∈ (𝐹𝐼)((𝑠𝐼) ∈ 𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
110100, 109sylibr 224 . . 3 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼))))
111 simpl2 1063 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → 𝐹:𝐴⟶Top)
112111, 25ffvelrnd 6316 . . . 4 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → (𝐹𝐼) ∈ Top)
113 eltop2 20690 . . . 4 ((𝐹𝐼) ∈ Top → (ran (𝑥𝑈 ↦ (𝑥𝐼)) ∈ (𝐹𝐼) ↔ ∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
114112, 113syl 17 . . 3 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → (ran (𝑥𝑈 ↦ (𝑥𝐼)) ∈ (𝐹𝐼) ↔ ∀𝑦 ∈ ran (𝑥𝑈 ↦ (𝑥𝐼))∃𝑧 ∈ (𝐹𝐼)(𝑦𝑧𝑧 ⊆ ran (𝑥𝑈 ↦ (𝑥𝐼)))))
115110, 114mpbird 247 . 2 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ran (𝑥𝑈 ↦ (𝑥𝐼)) ∈ (𝐹𝐼))
1168, 115eqeltrd 2698 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 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  wss 3555  ifcif 4058   cuni 4402  cmpt 4673  ran crn 5075  cres 5076  cima 5077   Fn wfn 5842  wf 5843  cfv 5847  Xcixp 7852  Fincfn 7899  topGenctg 16019  tcpt 16020  Topctop 20617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ixp 7853  df-topgen 16025  df-pt 16026  df-top 20621
This theorem is referenced by: (None)
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