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Theorem ptbasfi 23604
Description: The basis for the product topology can also be written as the set of finite intersections of "cylinder sets", the preimages of projections into one factor from open sets in the factor. (We have to add 𝑋 itself to the list because if 𝐴 is empty we get (fi‘∅) = ∅ while 𝐵 = {∅}.) (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypotheses
Ref Expression
ptbas.1 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
ptbasfi.2 𝑋 = X𝑛𝐴 (𝐹𝑛)
Assertion
Ref Expression
ptbasfi ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
Distinct variable groups:   𝑘,𝑛,𝑢,𝐵   𝑤,𝑔,𝑥,𝑦,𝑛,𝑘,𝑢,𝑧,𝐴   𝑔,𝐹,𝑘,𝑛,𝑢,𝑤,𝑥,𝑦,𝑧   𝑔,𝑋,𝑘,𝑢,𝑤,𝑥,𝑧   𝑔,𝑉,𝑘,𝑛,𝑢,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑤,𝑔)   𝑋(𝑦,𝑛)

Proof of Theorem ptbasfi
Dummy variables 𝑠 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptbas.1 . . . . 5 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
21elpt 23595 . . . 4 (𝑠𝐵 ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑦)))
3 df-3an 1088 . . . . . . . 8 (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)) ↔ (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦)) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)))
4 simprr 773 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))
5 disjdif2 4485 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) = ∅ → (𝐴𝑚) = 𝐴)
65raleqdv 3323 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) = ∅ → (∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ↔ ∀𝑦𝐴 (𝑦) = (𝐹𝑦)))
76biimpac 478 . . . . . . . . . . . . . . . 16 ((∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ∧ (𝐴𝑚) = ∅) → ∀𝑦𝐴 (𝑦) = (𝐹𝑦))
8 ixpeq2 8949 . . . . . . . . . . . . . . . 16 (∀𝑦𝐴 (𝑦) = (𝐹𝑦) → X𝑦𝐴 (𝑦) = X𝑦𝐴 (𝐹𝑦))
97, 8syl 17 . . . . . . . . . . . . . . 15 ((∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ∧ (𝐴𝑚) = ∅) → X𝑦𝐴 (𝑦) = X𝑦𝐴 (𝐹𝑦))
10 ptbasfi.2 . . . . . . . . . . . . . . . 16 𝑋 = X𝑛𝐴 (𝐹𝑛)
11 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑦 → (𝐹𝑛) = (𝐹𝑦))
1211unieqd 4924 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑦 (𝐹𝑛) = (𝐹𝑦))
1312cbvixpv 8953 . . . . . . . . . . . . . . . 16 X𝑛𝐴 (𝐹𝑛) = X𝑦𝐴 (𝐹𝑦)
1410, 13eqtri 2762 . . . . . . . . . . . . . . 15 𝑋 = X𝑦𝐴 (𝐹𝑦)
159, 14eqtr4di 2792 . . . . . . . . . . . . . 14 ((∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ∧ (𝐴𝑚) = ∅) → X𝑦𝐴 (𝑦) = 𝑋)
164, 15sylan 580 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) = ∅) → X𝑦𝐴 (𝑦) = 𝑋)
17 ssv 4019 . . . . . . . . . . . . . . . 16 𝑋 ⊆ V
18 iineq1 5013 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = 𝑛 ∈ ∅ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
19 0iin 5068 . . . . . . . . . . . . . . . . 17 𝑛 ∈ ∅ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V
2018, 19eqtrdi 2790 . . . . . . . . . . . . . . . 16 ((𝐴𝑚) = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V)
2117, 20sseqtrrid 4048 . . . . . . . . . . . . . . 15 ((𝐴𝑚) = ∅ → 𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
2221adantl 481 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) = ∅) → 𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
23 dfss2 3980 . . . . . . . . . . . . . 14 (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ↔ (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
2422, 23sylib 218 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) = ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
2516, 24eqtr4d 2777 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) = ∅) → X𝑦𝐴 (𝑦) = (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))))
26 simplll 775 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝐴𝑉𝐹:𝐴⟶Top))
27 inss1 4244 . . . . . . . . . . . . . . . . 17 (𝐴𝑚) ⊆ 𝐴
28 simpr 484 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚))
2927, 28sselid 3992 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛𝐴)
30 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → (𝑦) = (𝑛))
31 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → (𝐹𝑦) = (𝐹𝑛))
3230, 31eleq12d 2832 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑛 → ((𝑦) ∈ (𝐹𝑦) ↔ (𝑛) ∈ (𝐹𝑛)))
33 simprr 773 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
3433ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
3532, 34, 29rspcdva 3622 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑛) ∈ (𝐹𝑛))
3614ptpjpre1 23594 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑛𝐴 ∧ (𝑛) ∈ (𝐹𝑛))) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
3726, 29, 35, 36syl12anc 837 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
3837adantlr 715 . . . . . . . . . . . . . 14 ((((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
3938iineq2dv 5021 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = 𝑛 ∈ (𝐴𝑚)X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
40 simpr 484 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → (𝐴𝑚) ≠ ∅)
41 cnvimass 6101 . . . . . . . . . . . . . . . . . . . 20 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ dom (𝑤𝑋 ↦ (𝑤𝑛))
42 eqid 2734 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝑋 ↦ (𝑤𝑛)) = (𝑤𝑋 ↦ (𝑤𝑛))
4342dmmptss 6262 . . . . . . . . . . . . . . . . . . . 20 dom (𝑤𝑋 ↦ (𝑤𝑛)) ⊆ 𝑋
4441, 43sstri 4004 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋
4544, 14sseqtri 4031 . . . . . . . . . . . . . . . . . 18 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)
4645rgenw 3062 . . . . . . . . . . . . . . . . 17 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)
47 r19.2z 4500 . . . . . . . . . . . . . . . . 17 (((𝐴𝑚) ≠ ∅ ∧ ∀𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)) → ∃𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦))
4840, 46, 47sylancl 586 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∃𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦))
49 iinss 5060 . . . . . . . . . . . . . . . 16 (∃𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦))
5048, 49syl 17 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦))
5150, 14sseqtrrdi 4046 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋)
52 sseqin2 4230 . . . . . . . . . . . . . 14 ( 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋 ↔ (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
5351, 52sylib 218 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
5433ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
55 ssralv 4063 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) ⊆ 𝐴 → (∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) ∈ (𝐹𝑦)))
5627, 55ax-mp 5 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) ∈ (𝐹𝑦))
57 elssuni 4941 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ (𝐹𝑦))
58 iffalse 4539 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝐹𝑦))
5958sseq2d 4027 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦 = 𝑛 → ((𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (𝑦) ⊆ (𝐹𝑦)))
6057, 59syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦) ∈ (𝐹𝑦) → (¬ 𝑦 = 𝑛 → (𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
61 ssid 4017 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦) ⊆ (𝑦)
62 iftrue 4536 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑛))
6362, 30eqtr4d 2777 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑦))
6461, 63sseqtrrid 4048 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑛 → (𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6560, 64pm2.61d2 181 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6665ralrimivw 3147 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦) ∈ (𝐹𝑦) → ∀𝑛 ∈ (𝐴𝑚)(𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
67 ssiin 5059 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ ∀𝑛 ∈ (𝐴𝑚)(𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6866, 67sylibr 234 . . . . . . . . . . . . . . . . . . . 20 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6968adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → (𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
7062equcoms 2016 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑦 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑛))
71 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑦 → (𝑛) = (𝑦))
7270, 71eqtrd 2774 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑦 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑦))
7372sseq1d 4026 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑦 → (if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦) ↔ (𝑦) ⊆ (𝑦)))
7473rspcev 3621 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ⊆ (𝑦)) → ∃𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7561, 74mpan2 691 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝐴𝑚) → ∃𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
76 iinss 5060 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7775, 76syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝐴𝑚) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7877adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7969, 78eqssd 4012 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
8079ralimiaa 3079 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ (𝐴𝑚)(𝑦) ∈ (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
8154, 56, 803syl 18 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
82 eldifn 4141 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (𝐴𝑚) → ¬ 𝑦𝑚)
8382ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → ¬ 𝑦𝑚)
84 inss2 4245 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑚) ⊆ 𝑚
85 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚))
8684, 85sselid 3992 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛𝑚)
87 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑛 → (𝑦𝑚𝑛𝑚))
8886, 87syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑦 = 𝑛𝑦𝑚))
8983, 88mtod 198 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → ¬ 𝑦 = 𝑛)
9089, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝐹𝑦))
9190iineq2dv 5021 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = 𝑛 ∈ (𝐴𝑚) (𝐹𝑦))
92 iinconst 5006 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑚) ≠ ∅ → 𝑛 ∈ (𝐴𝑚) (𝐹𝑦) = (𝐹𝑦))
9392adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚) (𝐹𝑦) = (𝐹𝑦))
9491, 93eqtr2d 2775 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → (𝐹𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
95 eqeq1 2738 . . . . . . . . . . . . . . . . . . 19 ((𝑦) = (𝐹𝑦) → ((𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (𝐹𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
9694, 95syl5ibrcom 247 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → ((𝑦) = (𝐹𝑦) → (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
9796ralimdva 3164 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) ≠ ∅ → (∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
984, 97mpan9 506 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
99 inundif 4484 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) ∪ (𝐴𝑚)) = 𝐴
10099raleqi 3321 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ ((𝐴𝑚) ∪ (𝐴𝑚))(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ ∀𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
101 ralunb 4206 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ ((𝐴𝑚) ∪ (𝐴𝑚))(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
102100, 101bitr3i 277 . . . . . . . . . . . . . . . 16 (∀𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
10381, 98, 102sylanbrc 583 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∀𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
104 ixpeq2 8949 . . . . . . . . . . . . . . 15 (∀𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) → X𝑦𝐴 (𝑦) = X𝑦𝐴 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
105103, 104syl 17 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → X𝑦𝐴 (𝑦) = X𝑦𝐴 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
106 ixpiin 8962 . . . . . . . . . . . . . . 15 ((𝐴𝑚) ≠ ∅ → X𝑦𝐴 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = 𝑛 ∈ (𝐴𝑚)X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
107106adantl 481 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → X𝑦𝐴 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = 𝑛 ∈ (𝐴𝑚)X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
108105, 107eqtrd 2774 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → X𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
10939, 53, 1083eqtr4rd 2785 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → X𝑦𝐴 (𝑦) = (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))))
11025, 109pm2.61dane 3026 . . . . . . . . . . 11 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑦) = (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))))
111 ixpexg 8960 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑛𝐴 (𝐹𝑛) ∈ V → X𝑛𝐴 (𝐹𝑛) ∈ V)
112 fvex 6919 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹𝑛) ∈ V
113112uniex 7759 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹𝑛) ∈ V
114113a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛𝐴 (𝐹𝑛) ∈ V)
115111, 114mprg 3064 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑛𝐴 (𝐹𝑛) ∈ V
11610, 115eqeltri 2834 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 ∈ V
117116mptex 7242 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝑋 ↦ (𝑤𝑛)) ∈ V
118117cnvex 7947 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝑋 ↦ (𝑤𝑛)) ∈ V
119118imaex 7936 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ V
120119dfiin2 5038 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))}
121 inteq 4953 . . . . . . . . . . . . . . . . . 18 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅)
122120, 121eqtrid 2786 . . . . . . . . . . . . . . . . 17 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = ∅)
123 int0 4966 . . . . . . . . . . . . . . . . 17 ∅ = V
124122, 123eqtrdi 2790 . . . . . . . . . . . . . . . 16 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V)
125124ineq2d 4227 . . . . . . . . . . . . . . 15 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = (𝑋 ∩ V))
126 inv1 4403 . . . . . . . . . . . . . . 15 (𝑋 ∩ V) = 𝑋
127125, 126eqtrdi 2790 . . . . . . . . . . . . . 14 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
128127adantl 481 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
129 snex 5441 . . . . . . . . . . . . . . . . . 18 {𝑋} ∈ V
1301ptbas 23602 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)
1311, 10ptpjpre2 23603 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵)
132131ralrimivva 3199 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → ∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵)
133 eqid 2734 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
134133fmpox 8090 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵 ↔ (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)): 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝐵)
135132, 134sylib 218 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)): 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝐵)
136135frnd 6744 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ 𝐵)
137130, 136ssexd 5329 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ∈ V)
138 unexg 7761 . . . . . . . . . . . . . . . . . 18 (({𝑋} ∈ V ∧ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ∈ V) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V)
139129, 137, 138sylancr 587 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V)
140 ssfii 9456 . . . . . . . . . . . . . . . . 17 (({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
141139, 140syl 17 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
142141ad2antrr 726 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
143 ssun1 4187 . . . . . . . . . . . . . . . . 17 {𝑋} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
144116snss 4789 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ↔ {𝑋} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
145143, 144mpbir 231 . . . . . . . . . . . . . . . 16 𝑋 ∈ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
146145a1i 11 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → 𝑋 ∈ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
147142, 146sseldd 3995 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → 𝑋 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
148147adantr 480 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅) → 𝑋 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
149128, 148eqeltrd 2838 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
150139ad3antrrr 730 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V)
151 nfv 1911 . . . . . . . . . . . . . . . . . . . . . 22 𝑛(((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)))
152 nfcv 2902 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛𝐴
153 nfcv 2902 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛(𝐹𝑘)
154 nfixp1 8956 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑛X𝑛𝐴 (𝐹𝑛)
15510, 154nfcxfr 2900 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑛𝑋
156 nfcv 2902 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑛(𝑤𝑘)
157155, 156nfmpt 5254 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑛(𝑤𝑋 ↦ (𝑤𝑘))
158157nfcnv 5891 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑛(𝑤𝑋 ↦ (𝑤𝑘))
159 nfcv 2902 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑛𝑢
160158, 159nfima 6087 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)
161152, 153, 160nfmpo 7514 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
162161nfrn 5965 . . . . . . . . . . . . . . . . . . . . . . 23 𝑛ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
163162nfcri 2894 . . . . . . . . . . . . . . . . . . . . . 22 𝑛 𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
164 df-ov 7433 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))(𝑛)) = ((𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))‘⟨𝑛, (𝑛)⟩)
165119a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ V)
166 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑛 → (𝑤𝑘) = (𝑤𝑛))
167166mpteq2dv 5249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑛 → (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑛)))
168167cnveqd 5888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 = 𝑛(𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑛)))
169168imaeq1d 6078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑛 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ 𝑢))
170 imaeq2 6075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑢 = (𝑛) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
171169, 170sylan9eq 2794 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 = 𝑛𝑢 = (𝑛)) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
172 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
173171, 172, 133ovmpox 7585 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛𝐴 ∧ (𝑛) ∈ (𝐹𝑛) ∧ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ V) → (𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))(𝑛)) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
17429, 35, 165, 173syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))(𝑛)) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
175164, 174eqtr3id 2788 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))‘⟨𝑛, (𝑛)⟩) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
176135ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)): 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝐵)
177176ffnd 6737 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) Fn 𝑘𝐴 ({𝑘} × (𝐹𝑘)))
178 opeliunxp 5755 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑛, (𝑛)⟩ ∈ 𝑛𝐴 ({𝑛} × (𝐹𝑛)) ↔ (𝑛𝐴 ∧ (𝑛) ∈ (𝐹𝑛)))
17929, 35, 178sylanbrc 583 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ⟨𝑛, (𝑛)⟩ ∈ 𝑛𝐴 ({𝑛} × (𝐹𝑛)))
180 sneq 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → {𝑛} = {𝑘})
181 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
182180, 181xpeq12d 5719 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑘 → ({𝑛} × (𝐹𝑛)) = ({𝑘} × (𝐹𝑘)))
183182cbviunv 5044 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑛𝐴 ({𝑛} × (𝐹𝑛)) = 𝑘𝐴 ({𝑘} × (𝐹𝑘))
184179, 183eleqtrdi 2848 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ⟨𝑛, (𝑛)⟩ ∈ 𝑘𝐴 ({𝑘} × (𝐹𝑘)))
185 fnfvelrn 7099 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) Fn 𝑘𝐴 ({𝑘} × (𝐹𝑘)) ∧ ⟨𝑛, (𝑛)⟩ ∈ 𝑘𝐴 ({𝑘} × (𝐹𝑘))) → ((𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))‘⟨𝑛, (𝑛)⟩) ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
186177, 184, 185syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))‘⟨𝑛, (𝑛)⟩) ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
187175, 186eqeltrrd 2839 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
188 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) → (𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ↔ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
189187, 188syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) → 𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
190189ex 412 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → (𝑛 ∈ (𝐴𝑚) → (𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) → 𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
191151, 163, 190rexlimd 3263 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → (∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) → 𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
192191abssdv 4077 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ⊆ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
193 ssun2 4188 . . . . . . . . . . . . . . . . . . . 20 ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
194192, 193sstrdi 4007 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
195194adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
196 simpr 484 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅)
197 simplrl 777 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → 𝑚 ∈ Fin)
198 ssfi 9211 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ Fin ∧ (𝐴𝑚) ⊆ 𝑚) → (𝐴𝑚) ∈ Fin)
199197, 84, 198sylancl 586 . . . . . . . . . . . . . . . . . . 19 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → (𝐴𝑚) ∈ Fin)
200 abrexfi 9389 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑚) ∈ Fin → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ∈ Fin)
201199, 200syl 17 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ∈ Fin)
202 elfir 9452 . . . . . . . . . . . . . . . . . 18 ((({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V ∧ ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅ ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ∈ Fin)) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
203150, 195, 196, 201, 202syl13anc 1371 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
204120, 203eqeltrid 2842 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
205 elssuni 4941 . . . . . . . . . . . . . . . 16 ( 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
206204, 205syl 17 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
207 fiuni 9465 . . . . . . . . . . . . . . . . . 18 (({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
208139, 207syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
209116pwid 4626 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 ∈ 𝒫 𝑋
210209a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋 ∈ 𝒫 𝑋)
211210snssd 4813 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝒫 𝑋)
2121ptuni2 23599 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐵)
21310, 212eqtrid 2786 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋 = 𝐵)
214 eqimss2 4054 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑋 = 𝐵 𝐵𝑋)
215213, 214syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵𝑋)
216 sspwuni 5104 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵 ⊆ 𝒫 𝑋 𝐵𝑋)
217215, 216sylibr 234 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 𝑋)
218136, 217sstrd 4005 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ 𝒫 𝑋)
219211, 218unssd 4201 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋)
220 sspwuni 5104 . . . . . . . . . . . . . . . . . . 19 (({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋 ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝑋)
221219, 220sylib 218 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝑋)
222 elssuni 4941 . . . . . . . . . . . . . . . . . . 19 (𝑋 ∈ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑋 ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
223145, 222mp1i 13 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋 ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
224221, 223eqssd 4012 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) = 𝑋)
225208, 224eqtr3d 2776 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) = 𝑋)
226225ad3antrrr 730 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) = 𝑋)
227206, 226sseqtrd 4035 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋)
228227, 52sylib 218 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
229228, 204eqeltrd 2838 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
230149, 229pm2.61dane 3026 . . . . . . . . . . 11 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
231110, 230eqeltrd 2838 . . . . . . . . . 10 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
232231rexlimdvaa 3153 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) → (∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) → X𝑦𝐴 (𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
233232impr 454 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦)) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
2343, 233sylan2b 594 . . . . . . 7 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
235 eleq1 2826 . . . . . . 7 (𝑠 = X𝑦𝐴 (𝑦) → (𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ↔ X𝑦𝐴 (𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
236234, 235syl5ibrcom 247 . . . . . 6 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → (𝑠 = X𝑦𝐴 (𝑦) → 𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
237236expimpd 453 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → ((( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑦)) → 𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
238237exlimdv 1930 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → (∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑦)) → 𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
2392, 238biimtrid 242 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑠𝐵𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
240239ssrdv 4000 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
2411ptbasid 23598 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) ∈ 𝐵)
24210, 241eqeltrid 2842 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋𝐵)
243242snssd 4813 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝐵)
244243, 136unssd 4201 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝐵)
245 fiss 9461 . . . 4 ((𝐵 ∈ TopBases ∧ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝐵) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ⊆ (fi‘𝐵))
246130, 244, 245syl2anc 584 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ⊆ (fi‘𝐵))
2471ptbasin2 23601 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)
248246, 247sseqtrd 4035 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ⊆ 𝐵)
249240, 248eqssd 4012 1 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wne 2937  wral 3058  wrex 3067  Vcvv 3477  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  ifcif 4530  𝒫 cpw 4604  {csn 4630  cop 4636   cuni 4911   cint 4950   ciun 4995   ciin 4996  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  cima 5691   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  Xcixp 8935  Fincfn 8983  ficfi 9447  Topctop 22914  TopBasesctb 22967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-1o 8504  df-2o 8505  df-ixp 8936  df-en 8984  df-dom 8985  df-fin 8987  df-fi 9448  df-top 22915  df-bases 22968
This theorem is referenced by:  ptval2  23624  xkoptsub  23677  ptcmplem1  24075  prdsxmslem2  24557
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