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Theorem ptbasfi 22932
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 22923 . . . 4 (𝑠𝐵 ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑦)))
3 df-3an 1089 . . . . . . . 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 771 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))
5 disjdif2 4439 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) = ∅ → (𝐴𝑚) = 𝐴)
65raleqdv 3313 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) = ∅ → (∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ↔ ∀𝑦𝐴 (𝑦) = (𝐹𝑦)))
76biimpac 479 . . . . . . . . . . . . . . . 16 ((∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ∧ (𝐴𝑚) = ∅) → ∀𝑦𝐴 (𝑦) = (𝐹𝑦))
8 ixpeq2 8849 . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑦 → (𝐹𝑛) = (𝐹𝑦))
1211unieqd 4879 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑦 (𝐹𝑛) = (𝐹𝑦))
1312cbvixpv 8853 . . . . . . . . . . . . . . . 16 X𝑛𝐴 (𝐹𝑛) = X𝑦𝐴 (𝐹𝑦)
1410, 13eqtri 2764 . . . . . . . . . . . . . . 15 𝑋 = X𝑦𝐴 (𝐹𝑦)
159, 14eqtr4di 2794 . . . . . . . . . . . . . 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 3968 . . . . . . . . . . . . . . . 16 𝑋 ⊆ V
18 iineq1 4971 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = 𝑛 ∈ ∅ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
19 0iin 5024 . . . . . . . . . . . . . . . . 17 𝑛 ∈ ∅ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V
2018, 19eqtrdi 2792 . . . . . . . . . . . . . . . 16 ((𝐴𝑚) = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V)
2117, 20sseqtrrid 3997 . . . . . . . . . . . . . . 15 ((𝐴𝑚) = ∅ → 𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
2221adantl 482 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) = ∅) → 𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
23 df-ss 3927 . . . . . . . . . . . . . 14 (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ↔ (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
2422, 23sylib 217 . . . . . . . . . . . . 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 2779 . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝐴𝑉𝐹:𝐴⟶Top))
27 inss1 4188 . . . . . . . . . . . . . . . . 17 (𝐴𝑚) ⊆ 𝐴
28 simpr 485 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚))
2927, 28sselid 3942 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛𝐴)
30 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → (𝑦) = (𝑛))
31 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → (𝐹𝑦) = (𝐹𝑛))
3230, 31eleq12d 2832 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑛 → ((𝑦) ∈ (𝐹𝑦) ↔ (𝑛) ∈ (𝐹𝑛)))
33 simprr 771 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
3433ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
3532, 34, 29rspcdva 3582 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑛) ∈ (𝐹𝑛))
3614ptpjpre1 22922 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑛𝐴 ∧ (𝑛) ∈ (𝐹𝑛))) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
3726, 29, 35, 36syl12anc 835 . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . 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 4979 . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → (𝐴𝑚) ≠ ∅)
41 cnvimass 6033 . . . . . . . . . . . . . . . . . . . 20 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ dom (𝑤𝑋 ↦ (𝑤𝑛))
42 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝑋 ↦ (𝑤𝑛)) = (𝑤𝑋 ↦ (𝑤𝑛))
4342dmmptss 6193 . . . . . . . . . . . . . . . . . . . 20 dom (𝑤𝑋 ↦ (𝑤𝑛)) ⊆ 𝑋
4441, 43sstri 3953 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋
4544, 14sseqtri 3980 . . . . . . . . . . . . . . . . . 18 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)
4645rgenw 3068 . . . . . . . . . . . . . . . . 17 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)
47 r19.2z 4452 . . . . . . . . . . . . . . . . 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 5016 . . . . . . . . . . . . . . . 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 3995 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋)
52 sseqin2 4175 . . . . . . . . . . . . . 14 ( 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋 ↔ (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
5351, 52sylib 217 . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
55 ssralv 4010 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) ⊆ 𝐴 → (∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) ∈ (𝐹𝑦)))
5627, 55ax-mp 5 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) ∈ (𝐹𝑦))
57 elssuni 4898 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ (𝐹𝑦))
58 iffalse 4495 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝐹𝑦))
5958sseq2d 3976 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦 = 𝑛 → ((𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (𝑦) ⊆ (𝐹𝑦)))
6057, 59syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦) ∈ (𝐹𝑦) → (¬ 𝑦 = 𝑛 → (𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
61 ssid 3966 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦) ⊆ (𝑦)
62 iftrue 4492 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑛))
6362, 30eqtr4d 2779 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑦))
6461, 63sseqtrrid 3997 . . . . . . . . . . . . . . . . . . . . . . 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 5015 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ ∀𝑛 ∈ (𝐴𝑚)(𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6866, 67sylibr 233 . . . . . . . . . . . . . . . . . . . 20 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6968adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → (𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
7062equcoms 2023 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑦 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑛))
71 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑦 → (𝑛) = (𝑦))
7270, 71eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑦 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑦))
7372sseq1d 3975 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑦 → (if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦) ↔ (𝑦) ⊆ (𝑦)))
7473rspcev 3581 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ⊆ (𝑦)) → ∃𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7561, 74mpan2 689 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝐴𝑚) → ∃𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
76 iinss 5016 . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7969, 78eqssd 3961 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
8079ralimiaa 3085 . . . . . . . . . . . . . . . . 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 4087 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (𝐴𝑚) → ¬ 𝑦𝑚)
8382ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → ¬ 𝑦𝑚)
84 inss2 4189 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑚) ⊆ 𝑚
85 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚))
8684, 85sselid 3942 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛𝑚)
87 eleq1 2825 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑛 → (𝑦𝑚𝑛𝑚))
8886, 87syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑦 = 𝑛𝑦𝑚))
8983, 88mtod 197 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → ¬ 𝑦 = 𝑛)
9089, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝐹𝑦))
9190iineq2dv 4979 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = 𝑛 ∈ (𝐴𝑚) (𝐹𝑦))
92 iinconst 4964 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑚) ≠ ∅ → 𝑛 ∈ (𝐴𝑚) (𝐹𝑦) = (𝐹𝑦))
9392adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚) (𝐹𝑦) = (𝐹𝑦))
9491, 93eqtr2d 2777 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → (𝐹𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
95 eqeq1 2740 . . . . . . . . . . . . . . . . . . 19 ((𝑦) = (𝐹𝑦) → ((𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (𝐹𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
9694, 95syl5ibrcom 246 . . . . . . . . . . . . . . . . . 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 507 . . . . . . . . . . . . . . . 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 4438 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) ∪ (𝐴𝑚)) = 𝐴
10099raleqi 3311 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ ((𝐴𝑚) ∪ (𝐴𝑚))(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ ∀𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
101 ralunb 4151 . . . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . . 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 8849 . . . . . . . . . . . . . . 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 8862 . . . . . . . . . . . . . . 15 ((𝐴𝑚) ≠ ∅ → X𝑦𝐴 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = 𝑛 ∈ (𝐴𝑚)X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
107106adantl 482 . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . 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 2787 . . . . . . . . . . . 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 3032 . . . . . . . . . . 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 8860 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑛𝐴 (𝐹𝑛) ∈ V → X𝑛𝐴 (𝐹𝑛) ∈ V)
112 fvex 6855 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹𝑛) ∈ V
113112uniex 7678 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹𝑛) ∈ V
114113a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛𝐴 (𝐹𝑛) ∈ V)
115111, 114mprg 3070 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑛𝐴 (𝐹𝑛) ∈ V
11610, 115eqeltri 2834 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 ∈ V
117116mptex 7173 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝑋 ↦ (𝑤𝑛)) ∈ V
118117cnvex 7862 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝑋 ↦ (𝑤𝑛)) ∈ V
119118imaex 7853 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ V
120119dfiin2 4994 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))}
121 inteq 4910 . . . . . . . . . . . . . . . . . 18 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅)
122120, 121eqtrid 2788 . . . . . . . . . . . . . . . . 17 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = ∅)
123 int0 4923 . . . . . . . . . . . . . . . . 17 ∅ = V
124122, 123eqtrdi 2792 . . . . . . . . . . . . . . . 16 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V)
125124ineq2d 4172 . . . . . . . . . . . . . . 15 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = (𝑋 ∩ V))
126 inv1 4354 . . . . . . . . . . . . . . 15 (𝑋 ∩ V) = 𝑋
127125, 126eqtrdi 2792 . . . . . . . . . . . . . 14 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
128127adantl 482 . . . . . . . . . . . . 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 5388 . . . . . . . . . . . . . . . . . 18 {𝑋} ∈ V
1301ptbas 22930 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)
1311, 10ptpjpre2 22931 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵)
132131ralrimivva 3197 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → ∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵)
133 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
134133fmpox 7999 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵 ↔ (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)): 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝐵)
135132, 134sylib 217 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)): 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝐵)
136135frnd 6676 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ 𝐵)
137130, 136ssexd 5281 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ∈ V)
138 unexg 7683 . . . . . . . . . . . . . . . . . 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 9355 . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . 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 4132 . . . . . . . . . . . . . . . . 17 {𝑋} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
144116snss 4746 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ↔ {𝑋} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
145143, 144mpbir 230 . . . . . . . . . . . . . . . 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 3945 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . 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 1917 . . . . . . . . . . . . . . . . . . . . . 22 𝑛(((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)))
152 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛𝐴
153 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛(𝐹𝑘)
154 nfixp1 8856 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑛X𝑛𝐴 (𝐹𝑛)
15510, 154nfcxfr 2905 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑛𝑋
156 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑛(𝑤𝑘)
157155, 156nfmpt 5212 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑛(𝑤𝑋 ↦ (𝑤𝑘))
158157nfcnv 5834 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑛(𝑤𝑋 ↦ (𝑤𝑘))
159 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑛𝑢
160158, 159nfima 6021 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)
161152, 153, 160nfmpo 7439 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
162161nfrn 5907 . . . . . . . . . . . . . . . . . . . . . . 23 𝑛ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
163162nfcri 2894 . . . . . . . . . . . . . . . . . . . . . 22 𝑛 𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
164 df-ov 7360 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑛 → (𝑤𝑘) = (𝑤𝑛))
167166mpteq2dv 5207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑛 → (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑛)))
168167cnveqd 5831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 = 𝑛(𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑛)))
169168imaeq1d 6012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑛 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ 𝑢))
170 imaeq2 6009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑢 = (𝑛) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
171169, 170sylan9eq 2796 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 = 𝑛𝑢 = (𝑛)) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
172 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
173171, 172, 133ovmpox 7508 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛𝐴 ∧ (𝑛) ∈ (𝐹𝑛) ∧ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ V) → (𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))(𝑛)) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
17429, 35, 165, 173syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6669 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5699 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → {𝑛} = {𝑘})
181 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
182180, 181xpeq12d 5664 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑘 → ({𝑛} × (𝐹𝑛)) = ({𝑘} × (𝐹𝑘)))
183182cbviunv 5000 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . 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 3249 . . . . . . . . . . . . . . . . . . . . 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 4025 . . . . . . . . . . . . . . . . . . . 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 4133 . . . . . . . . . . . . . . . . . . . 20 ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
194192, 193sstrdi 3956 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . . 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 9117 . . . . . . . . . . . . . . . . . . . 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 9296 . . . . . . . . . . . . . . . . . . 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 9351 . . . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . . . 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 4898 . . . . . . . . . . . . . . . 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 9364 . . . . . . . . . . . . . . . . . 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 4582 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 ∈ 𝒫 𝑋
210209a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋 ∈ 𝒫 𝑋)
211210snssd 4769 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝒫 𝑋)
2121ptuni2 22927 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐵)
21310, 212eqtrid 2788 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋 = 𝐵)
214 eqimss2 4001 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑋 = 𝐵 𝐵𝑋)
215213, 214syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵𝑋)
216 sspwuni 5060 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵 ⊆ 𝒫 𝑋 𝐵𝑋)
217215, 216sylibr 233 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 𝑋)
218136, 217sstrd 3954 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ 𝒫 𝑋)
219211, 218unssd 4146 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋)
220 sspwuni 5060 . . . . . . . . . . . . . . . . . . 19 (({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋 ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝑋)
221219, 220sylib 217 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝑋)
222 elssuni 4898 . . . . . . . . . . . . . . . . . . 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 3961 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) = 𝑋)
225208, 224eqtr3d 2778 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) = 𝑋)
226225ad3antrrr 728 . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . 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 3032 . . . . . . . . . . 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 455 . . . . . . . 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 2825 . . . . . . 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 246 . . . . . 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 454 . . . . 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 1936 . . . 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 241 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑠𝐵𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
240239ssrdv 3950 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
2411ptbasid 22926 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) ∈ 𝐵)
24210, 241eqeltrid 2842 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋𝐵)
243242snssd 4769 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝐵)
244243, 136unssd 4146 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝐵)
245 fiss 9360 . . . 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 22929 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)
248246, 247sseqtrd 3984 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ⊆ 𝐵)
249240, 248eqssd 3961 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 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  ifcif 4486  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   cint 4907   ciun 4954   ciin 4955  cmpt 5188   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  Xcixp 8835  Fincfn 8883  ficfi 9346  Topctop 22242  TopBasesctb 22295
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-3or 1088  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-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  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-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  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-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-ixp 8836  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-top 22243  df-bases 22296
This theorem is referenced by:  ptval2  22952  xkoptsub  23005  ptcmplem1  23403  prdsxmslem2  23885
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