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Theorem ptbasfi 22227
 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 22218 . . . 4 (𝑠𝐵 ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑦)))
3 df-3an 1086 . . . . . . . 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 772 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))
5 disjdif2 4389 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) = ∅ → (𝐴𝑚) = 𝐴)
65raleqdv 3365 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) = ∅ → (∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ↔ ∀𝑦𝐴 (𝑦) = (𝐹𝑦)))
76biimpac 482 . . . . . . . . . . . . . . . 16 ((∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ∧ (𝐴𝑚) = ∅) → ∀𝑦𝐴 (𝑦) = (𝐹𝑦))
8 ixpeq2 8476 . . . . . . . . . . . . . . . 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 6655 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑦 → (𝐹𝑛) = (𝐹𝑦))
1211unieqd 4818 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑦 (𝐹𝑛) = (𝐹𝑦))
1312cbvixpv 8480 . . . . . . . . . . . . . . . 16 X𝑛𝐴 (𝐹𝑛) = X𝑦𝐴 (𝐹𝑦)
1410, 13eqtri 2821 . . . . . . . . . . . . . . 15 𝑋 = X𝑦𝐴 (𝐹𝑦)
159, 14eqtr4di 2851 . . . . . . . . . . . . . 14 ((∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ∧ (𝐴𝑚) = ∅) → X𝑦𝐴 (𝑦) = 𝑋)
164, 15sylan 583 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) = ∅) → X𝑦𝐴 (𝑦) = 𝑋)
17 ssv 3941 . . . . . . . . . . . . . . . 16 𝑋 ⊆ V
18 iineq1 4902 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = 𝑛 ∈ ∅ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
19 0iin 4954 . . . . . . . . . . . . . . . . 17 𝑛 ∈ ∅ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V
2018, 19eqtrdi 2849 . . . . . . . . . . . . . . . 16 ((𝐴𝑚) = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V)
2117, 20sseqtrrid 3970 . . . . . . . . . . . . . . 15 ((𝐴𝑚) = ∅ → 𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
2221adantl 485 . . . . . . . . . . . . . 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 3900 . . . . . . . . . . . . . 14 (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ↔ (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
2422, 23sylib 221 . . . . . . . . . . . . 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 2836 . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝐴𝑉𝐹:𝐴⟶Top))
27 inss1 4158 . . . . . . . . . . . . . . . . 17 (𝐴𝑚) ⊆ 𝐴
28 simpr 488 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚))
2927, 28sseldi 3915 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛𝐴)
30 fveq2 6655 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → (𝑦) = (𝑛))
31 fveq2 6655 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → (𝐹𝑦) = (𝐹𝑛))
3230, 31eleq12d 2884 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑛 → ((𝑦) ∈ (𝐹𝑦) ↔ (𝑛) ∈ (𝐹𝑛)))
33 simprr 772 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
3433ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
3532, 34, 29rspcdva 3574 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑛) ∈ (𝐹𝑛))
3614ptpjpre1 22217 . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . 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 4910 . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → (𝐴𝑚) ≠ ∅)
41 cnvimass 5920 . . . . . . . . . . . . . . . . . . . 20 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ dom (𝑤𝑋 ↦ (𝑤𝑛))
42 eqid 2798 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝑋 ↦ (𝑤𝑛)) = (𝑤𝑋 ↦ (𝑤𝑛))
4342dmmptss 6067 . . . . . . . . . . . . . . . . . . . 20 dom (𝑤𝑋 ↦ (𝑤𝑛)) ⊆ 𝑋
4441, 43sstri 3926 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋
4544, 14sseqtri 3953 . . . . . . . . . . . . . . . . . 18 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)
4645rgenw 3118 . . . . . . . . . . . . . . . . 17 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)
47 r19.2z 4401 . . . . . . . . . . . . . . . . 17 (((𝐴𝑚) ≠ ∅ ∧ ∀𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)) → ∃𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦))
4840, 46, 47sylancl 589 . . . . . . . . . . . . . . . 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 4947 . . . . . . . . . . . . . . . 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 3968 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋)
52 sseqin2 4145 . . . . . . . . . . . . . 14 ( 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋 ↔ (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
5351, 52sylib 221 . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
55 ssralv 3983 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) ⊆ 𝐴 → (∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) ∈ (𝐹𝑦)))
5627, 55ax-mp 5 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) ∈ (𝐹𝑦))
57 elssuni 4834 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ (𝐹𝑦))
58 iffalse 4437 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝐹𝑦))
5958sseq2d 3949 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦 = 𝑛 → ((𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (𝑦) ⊆ (𝐹𝑦)))
6057, 59syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦) ∈ (𝐹𝑦) → (¬ 𝑦 = 𝑛 → (𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
61 ssid 3939 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦) ⊆ (𝑦)
62 iftrue 4434 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑛))
6362, 30eqtr4d 2836 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑦))
6461, 63sseqtrrid 3970 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑛 → (𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6560, 64pm2.61d2 184 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6665ralrimivw 3150 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦) ∈ (𝐹𝑦) → ∀𝑛 ∈ (𝐴𝑚)(𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
67 ssiin 4946 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ ∀𝑛 ∈ (𝐴𝑚)(𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6866, 67sylibr 237 . . . . . . . . . . . . . . . . . . . 20 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6968adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → (𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
7062equcoms 2027 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑦 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑛))
71 fveq2 6655 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑦 → (𝑛) = (𝑦))
7270, 71eqtrd 2833 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑦 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑦))
7372sseq1d 3948 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑦 → (if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦) ↔ (𝑦) ⊆ (𝑦)))
7473rspcev 3572 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ⊆ (𝑦)) → ∃𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7561, 74mpan2 690 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝐴𝑚) → ∃𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
76 iinss 4947 . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7969, 78eqssd 3934 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
8079ralimiaa 3127 . . . . . . . . . . . . . . . . 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 4058 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (𝐴𝑚) → ¬ 𝑦𝑚)
8382ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → ¬ 𝑦𝑚)
84 inss2 4159 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑚) ⊆ 𝑚
85 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚))
8684, 85sseldi 3915 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛𝑚)
87 eleq1 2877 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑛 → (𝑦𝑚𝑛𝑚))
8886, 87syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑦 = 𝑛𝑦𝑚))
8983, 88mtod 201 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → ¬ 𝑦 = 𝑛)
9089, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝐹𝑦))
9190iineq2dv 4910 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = 𝑛 ∈ (𝐴𝑚) (𝐹𝑦))
92 iinconst 4895 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑚) ≠ ∅ → 𝑛 ∈ (𝐴𝑚) (𝐹𝑦) = (𝐹𝑦))
9392adantr 484 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚) (𝐹𝑦) = (𝐹𝑦))
9491, 93eqtr2d 2834 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → (𝐹𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
95 eqeq1 2802 . . . . . . . . . . . . . . . . . . 19 ((𝑦) = (𝐹𝑦) → ((𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (𝐹𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
9694, 95syl5ibrcom 250 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → ((𝑦) = (𝐹𝑦) → (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
9796ralimdva 3144 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) ≠ ∅ → (∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
984, 97mpan9 510 . . . . . . . . . . . . . . . 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 4388 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) ∪ (𝐴𝑚)) = 𝐴
10099raleqi 3363 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ ((𝐴𝑚) ∪ (𝐴𝑚))(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ ∀𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
101 ralunb 4121 . . . . . . . . . . . . . . . . 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 280 . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . 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 8476 . . . . . . . . . . . . . . 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 8489 . . . . . . . . . . . . . . 15 ((𝐴𝑚) ≠ ∅ → X𝑦𝐴 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = 𝑛 ∈ (𝐴𝑚)X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
107106adantl 485 . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . 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 2844 . . . . . . . . . . . 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 3074 . . . . . . . . . . 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 8487 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑛𝐴 (𝐹𝑛) ∈ V → X𝑛𝐴 (𝐹𝑛) ∈ V)
112 fvex 6668 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹𝑛) ∈ V
113112uniex 7460 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹𝑛) ∈ V
114113a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛𝐴 (𝐹𝑛) ∈ V)
115111, 114mprg 3120 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑛𝐴 (𝐹𝑛) ∈ V
11610, 115eqeltri 2886 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 ∈ V
117116mptex 6973 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝑋 ↦ (𝑤𝑛)) ∈ V
118117cnvex 7625 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝑋 ↦ (𝑤𝑛)) ∈ V
119118imaex 7616 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ V
120119dfiin2 4925 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))}
121 inteq 4845 . . . . . . . . . . . . . . . . . 18 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅)
122120, 121syl5eq 2845 . . . . . . . . . . . . . . . . 17 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = ∅)
123 int0 4856 . . . . . . . . . . . . . . . . 17 ∅ = V
124122, 123eqtrdi 2849 . . . . . . . . . . . . . . . 16 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V)
125124ineq2d 4142 . . . . . . . . . . . . . . 15 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = (𝑋 ∩ V))
126 inv1 4305 . . . . . . . . . . . . . . 15 (𝑋 ∩ V) = 𝑋
127125, 126eqtrdi 2849 . . . . . . . . . . . . . 14 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
128127adantl 485 . . . . . . . . . . . . 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 5301 . . . . . . . . . . . . . . . . . 18 {𝑋} ∈ V
1301ptbas 22225 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)
1311, 10ptpjpre2 22226 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵)
132131ralrimivva 3156 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → ∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵)
133 eqid 2798 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
134133fmpox 7760 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵 ↔ (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)): 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝐵)
135132, 134sylib 221 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)): 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝐵)
136135frnd 6502 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ 𝐵)
137130, 136ssexd 5196 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ∈ V)
138 unexg 7465 . . . . . . . . . . . . . . . . . 18 (({𝑋} ∈ V ∧ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ∈ V) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V)
139129, 137, 138sylancr 590 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V)
140 ssfii 8885 . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . 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 4102 . . . . . . . . . . . . . . . . 17 {𝑋} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
144116snss 4682 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ↔ {𝑋} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
145143, 144mpbir 234 . . . . . . . . . . . . . . . 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 3918 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 2890 . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . . . 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 1915 . . . . . . . . . . . . . . . . . . . . . 22 𝑛(((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)))
152 nfcv 2955 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛𝐴
153 nfcv 2955 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛(𝐹𝑘)
154 nfixp1 8483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑛X𝑛𝐴 (𝐹𝑛)
15510, 154nfcxfr 2953 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑛𝑋
156 nfcv 2955 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑛(𝑤𝑘)
157155, 156nfmpt 5131 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑛(𝑤𝑋 ↦ (𝑤𝑘))
158157nfcnv 5717 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑛(𝑤𝑋 ↦ (𝑤𝑘))
159 nfcv 2955 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑛𝑢
160158, 159nfima 5908 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)
161152, 153, 160nfmpo 7225 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
162161nfrn 5792 . . . . . . . . . . . . . . . . . . . . . . 23 𝑛ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
163162nfcri 2943 . . . . . . . . . . . . . . . . . . . . . 22 𝑛 𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
164 df-ov 7148 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑛 → (𝑤𝑘) = (𝑤𝑛))
167166mpteq2dv 5130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑛 → (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑛)))
168167cnveqd 5714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 = 𝑛(𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑛)))
169168imaeq1d 5899 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑛 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ 𝑢))
170 imaeq2 5896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑢 = (𝑛) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
171169, 170sylan9eq 2853 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 = 𝑛𝑢 = (𝑛)) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
172 fveq2 6655 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
173171, 172, 133ovmpox 7293 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛𝐴 ∧ (𝑛) ∈ (𝐹𝑛) ∧ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ V) → (𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))(𝑛)) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
17429, 35, 165, 173syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . 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, 174syl5eqr 2847 . . . . . . . . . . . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6496 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑛, (𝑛)⟩ ∈ 𝑛𝐴 ({𝑛} × (𝐹𝑛)) ↔ (𝑛𝐴 ∧ (𝑛) ∈ (𝐹𝑛)))
17929, 35, 178sylanbrc 586 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ⟨𝑛, (𝑛)⟩ ∈ 𝑛𝐴 ({𝑛} × (𝐹𝑛)))
180 sneq 4538 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → {𝑛} = {𝑘})
181 fveq2 6655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
182180, 181xpeq12d 5554 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑘 → ({𝑛} × (𝐹𝑛)) = ({𝑘} × (𝐹𝑘)))
183182cbviunv 4931 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑛𝐴 ({𝑛} × (𝐹𝑛)) = 𝑘𝐴 ({𝑘} × (𝐹𝑘))
184179, 183eleqtrdi 2900 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ⟨𝑛, (𝑛)⟩ ∈ 𝑘𝐴 ({𝑘} × (𝐹𝑘)))
185 fnfvelrn 6835 . . . . . . . . . . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . . . . . . . . . . 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 2891 . . . . . . . . . . . . . . . . . . . . . . . 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 2877 . . . . . . . . . . . . . . . . . . . . . . . 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 250 . . . . . . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . . . . . . . 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 3277 . . . . . . . . . . . . . . . . . . . . 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 3998 . . . . . . . . . . . . . . . . . . . 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 4103 . . . . . . . . . . . . . . . . . . . 20 ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
194192, 193sstrdi 3929 . . . . . . . . . . . . . . . . . . 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 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))} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
196 simpr 488 . . . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . . 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 8740 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ Fin ∧ (𝐴𝑚) ⊆ 𝑚) → (𝐴𝑚) ∈ Fin)
199197, 84, 198sylancl 589 . . . . . . . . . . . . . . . . . . 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 8826 . . . . . . . . . . . . . . . . . . 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 8881 . . . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . . . 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 2894 . . . . . . . . . . . . . . . 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 4834 . . . . . . . . . . . . . . . 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 8894 . . . . . . . . . . . . . . . . . 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 4524 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 ∈ 𝒫 𝑋
210209a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋 ∈ 𝒫 𝑋)
211210snssd 4705 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝒫 𝑋)
2121ptuni2 22222 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐵)
21310, 212syl5eq 2845 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋 = 𝐵)
214 eqimss2 3974 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑋 = 𝐵 𝐵𝑋)
215213, 214syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵𝑋)
216 sspwuni 4989 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵 ⊆ 𝒫 𝑋 𝐵𝑋)
217215, 216sylibr 237 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 𝑋)
218136, 217sstrd 3927 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ 𝒫 𝑋)
219211, 218unssd 4116 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋)
220 sspwuni 4989 . . . . . . . . . . . . . . . . . . 19 (({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋 ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝑋)
221219, 220sylib 221 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝑋)
222 elssuni 4834 . . . . . . . . . . . . . . . . . . 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 3934 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) = 𝑋)
225208, 224eqtr3d 2835 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) = 𝑋)
226225ad3antrrr 729 . . . . . . . . . . . . . . 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 3957 . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . 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 2890 . . . . . . . . . . . 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 3074 . . . . . . . . . . 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 2890 . . . . . . . . . 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 3245 . . . . . . . . 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 458 . . . . . . . 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 596 . . . . . . 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 2877 . . . . . . 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 250 . . . . . 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 457 . . . . 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 1934 . . . 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, 238syl5bi 245 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑠𝐵𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
240239ssrdv 3923 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
2411ptbasid 22221 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) ∈ 𝐵)
24210, 241eqeltrid 2894 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋𝐵)
243242snssd 4705 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝐵)
244243, 136unssd 4116 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝐵)
245 fiss 8890 . . . 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 587 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ⊆ (fi‘𝐵))
2471ptbasin2 22224 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)
248246, 247sseqtrd 3957 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ⊆ 𝐵)
249240, 248eqssd 3934 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 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3442   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4246  ifcif 4428  𝒫 cpw 4500  {csn 4528  ⟨cop 4534  ∪ cuni 4804  ∩ cint 4842  ∪ ciun 4885  ∩ ciin 4886   ↦ cmpt 5114   × cxp 5521  ◡ccnv 5522  dom cdm 5523  ran crn 5524   “ cima 5526   Fn wfn 6327  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145   ∈ cmpo 7147  Xcixp 8462  Fincfn 8510  ficfi 8876  Topctop 21539  TopBasesctb 21591 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-oadd 8107  df-er 8290  df-ixp 8463  df-en 8511  df-dom 8512  df-fin 8514  df-fi 8877  df-top 21540  df-bases 21592 This theorem is referenced by:  ptval2  22247  xkoptsub  22300  ptcmplem1  22698  prdsxmslem2  23177
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