Metamath Proof Explorer


Theorem ptbasfi

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)

Ref Expression
Hypotheses ptbas.1 𝐵 = { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑔𝑦 ) ∈ ( 𝐹𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑧 ) ( 𝑔𝑦 ) = ( 𝐹𝑦 ) ) ∧ 𝑥 = X 𝑦𝐴 ( 𝑔𝑦 ) ) }
ptbasfi.2 𝑋 = X 𝑛𝐴 ( 𝐹𝑛 )
Assertion ptbasfi ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → 𝐵 = ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 ptbas.1 𝐵 = { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑔𝑦 ) ∈ ( 𝐹𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑧 ) ( 𝑔𝑦 ) = ( 𝐹𝑦 ) ) ∧ 𝑥 = X 𝑦𝐴 ( 𝑔𝑦 ) ) }
2 ptbasfi.2 𝑋 = X 𝑛𝐴 ( 𝐹𝑛 )
3 1 elpt ( 𝑠𝐵 ↔ ∃ ( ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ∧ 𝑠 = X 𝑦𝐴 ( 𝑦 ) ) )
4 df-3an ( ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ↔ ( ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) )
5 simprr ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) → ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) )
6 disjdif2 ( ( 𝐴𝑚 ) = ∅ → ( 𝐴𝑚 ) = 𝐴 )
7 6 raleqdv ( ( 𝐴𝑚 ) = ∅ → ( ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ↔ ∀ 𝑦𝐴 ( 𝑦 ) = ( 𝐹𝑦 ) ) )
8 7 biimpac ( ( ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ∧ ( 𝐴𝑚 ) = ∅ ) → ∀ 𝑦𝐴 ( 𝑦 ) = ( 𝐹𝑦 ) )
9 ixpeq2 ( ∀ 𝑦𝐴 ( 𝑦 ) = ( 𝐹𝑦 ) → X 𝑦𝐴 ( 𝑦 ) = X 𝑦𝐴 ( 𝐹𝑦 ) )
10 8 9 syl ( ( ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ∧ ( 𝐴𝑚 ) = ∅ ) → X 𝑦𝐴 ( 𝑦 ) = X 𝑦𝐴 ( 𝐹𝑦 ) )
11 fveq2 ( 𝑛 = 𝑦 → ( 𝐹𝑛 ) = ( 𝐹𝑦 ) )
12 11 unieqd ( 𝑛 = 𝑦 ( 𝐹𝑛 ) = ( 𝐹𝑦 ) )
13 12 cbvixpv X 𝑛𝐴 ( 𝐹𝑛 ) = X 𝑦𝐴 ( 𝐹𝑦 )
14 2 13 eqtri 𝑋 = X 𝑦𝐴 ( 𝐹𝑦 )
15 10 14 eqtr4di ( ( ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ∧ ( 𝐴𝑚 ) = ∅ ) → X 𝑦𝐴 ( 𝑦 ) = 𝑋 )
16 5 15 sylan ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) = ∅ ) → X 𝑦𝐴 ( 𝑦 ) = 𝑋 )
17 ssv 𝑋 ⊆ V
18 iineq1 ( ( 𝐴𝑚 ) = ∅ → 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) = 𝑛 ∈ ∅ ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) )
19 0iin 𝑛 ∈ ∅ ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) = V
20 18 19 eqtrdi ( ( 𝐴𝑚 ) = ∅ → 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) = V )
21 17 20 sseqtrrid ( ( 𝐴𝑚 ) = ∅ → 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) )
22 21 adantl ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) = ∅ ) → 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) )
23 df-ss ( 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ↔ ( 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ) = 𝑋 )
24 22 23 sylib ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) = ∅ ) → ( 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ) = 𝑋 )
25 16 24 eqtr4d ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) = ∅ ) → X 𝑦𝐴 ( 𝑦 ) = ( 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ) )
26 simplll ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) )
27 inss1 ( 𝐴𝑚 ) ⊆ 𝐴
28 simpr ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → 𝑛 ∈ ( 𝐴𝑚 ) )
29 27 28 sselid ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → 𝑛𝐴 )
30 fveq2 ( 𝑦 = 𝑛 → ( 𝑦 ) = ( 𝑛 ) )
31 fveq2 ( 𝑦 = 𝑛 → ( 𝐹𝑦 ) = ( 𝐹𝑛 ) )
32 30 31 eleq12d ( 𝑦 = 𝑛 → ( ( 𝑦 ) ∈ ( 𝐹𝑦 ) ↔ ( 𝑛 ) ∈ ( 𝐹𝑛 ) ) )
33 simprr ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) → ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) )
34 33 ad2antrr ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) )
35 32 34 29 rspcdva ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ( 𝑛 ) ∈ ( 𝐹𝑛 ) )
36 14 ptpjpre1 ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝑛𝐴 ∧ ( 𝑛 ) ∈ ( 𝐹𝑛 ) ) ) → ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) = X 𝑦𝐴 if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
37 26 29 35 36 syl12anc ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) = X 𝑦𝐴 if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
38 37 adantlr ( ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) = X 𝑦𝐴 if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
39 38 iineq2dv ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) → 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) = 𝑛 ∈ ( 𝐴𝑚 ) X 𝑦𝐴 if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
40 simpr ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) → ( 𝐴𝑚 ) ≠ ∅ )
41 cnvimass ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ dom ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) )
42 eqid ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) = ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) )
43 42 dmmptss dom ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) ⊆ 𝑋
44 41 43 sstri ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ 𝑋
45 44 14 sseqtri ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ X 𝑦𝐴 ( 𝐹𝑦 )
46 45 rgenw 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ X 𝑦𝐴 ( 𝐹𝑦 )
47 r19.2z ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ X 𝑦𝐴 ( 𝐹𝑦 ) ) → ∃ 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ X 𝑦𝐴 ( 𝐹𝑦 ) )
48 40 46 47 sylancl ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) → ∃ 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ X 𝑦𝐴 ( 𝐹𝑦 ) )
49 iinss ( ∃ 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ X 𝑦𝐴 ( 𝐹𝑦 ) → 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ X 𝑦𝐴 ( 𝐹𝑦 ) )
50 48 49 syl ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) → 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ X 𝑦𝐴 ( 𝐹𝑦 ) )
51 50 14 sseqtrrdi ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) → 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ 𝑋 )
52 sseqin2 ( 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ⊆ 𝑋 ↔ ( 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ) = 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) )
53 51 52 sylib ( ( ( ( ( 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 ) ) )
54 33 ad2antrr ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) → ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) )
55 ssralv ( ( 𝐴𝑚 ) ⊆ 𝐴 → ( ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) → ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) )
56 27 55 ax-mp ( ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) → ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) ∈ ( 𝐹𝑦 ) )
57 elssuni ( ( 𝑦 ) ∈ ( 𝐹𝑦 ) → ( 𝑦 ) ⊆ ( 𝐹𝑦 ) )
58 iffalse ( ¬ 𝑦 = 𝑛 → if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) = ( 𝐹𝑦 ) )
59 58 sseq2d ( ¬ 𝑦 = 𝑛 → ( ( 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ↔ ( 𝑦 ) ⊆ ( 𝐹𝑦 ) ) )
60 57 59 syl5ibrcom ( ( 𝑦 ) ∈ ( 𝐹𝑦 ) → ( ¬ 𝑦 = 𝑛 → ( 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ) )
61 ssid ( 𝑦 ) ⊆ ( 𝑦 )
62 iftrue ( 𝑦 = 𝑛 → if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) = ( 𝑛 ) )
63 62 30 eqtr4d ( 𝑦 = 𝑛 → if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) = ( 𝑦 ) )
64 61 63 sseqtrrid ( 𝑦 = 𝑛 → ( 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
65 60 64 pm2.61d2 ( ( 𝑦 ) ∈ ( 𝐹𝑦 ) → ( 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
66 65 ralrimivw ( ( 𝑦 ) ∈ ( 𝐹𝑦 ) → ∀ 𝑛 ∈ ( 𝐴𝑚 ) ( 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
67 ssiin ( ( 𝑦 ) ⊆ 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ↔ ∀ 𝑛 ∈ ( 𝐴𝑚 ) ( 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
68 66 67 sylibr ( ( 𝑦 ) ∈ ( 𝐹𝑦 ) → ( 𝑦 ) ⊆ 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
69 68 adantl ( ( 𝑦 ∈ ( 𝐴𝑚 ) ∧ ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) → ( 𝑦 ) ⊆ 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
70 62 equcoms ( 𝑛 = 𝑦 → if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) = ( 𝑛 ) )
71 fveq2 ( 𝑛 = 𝑦 → ( 𝑛 ) = ( 𝑦 ) )
72 70 71 eqtrd ( 𝑛 = 𝑦 → if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) = ( 𝑦 ) )
73 72 sseq1d ( 𝑛 = 𝑦 → ( if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ⊆ ( 𝑦 ) ↔ ( 𝑦 ) ⊆ ( 𝑦 ) ) )
74 73 rspcev ( ( 𝑦 ∈ ( 𝐴𝑚 ) ∧ ( 𝑦 ) ⊆ ( 𝑦 ) ) → ∃ 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ⊆ ( 𝑦 ) )
75 61 74 mpan2 ( 𝑦 ∈ ( 𝐴𝑚 ) → ∃ 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ⊆ ( 𝑦 ) )
76 iinss ( ∃ 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ⊆ ( 𝑦 ) → 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ⊆ ( 𝑦 ) )
77 75 76 syl ( 𝑦 ∈ ( 𝐴𝑚 ) → 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ⊆ ( 𝑦 ) )
78 77 adantr ( ( 𝑦 ∈ ( 𝐴𝑚 ) ∧ ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) → 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ⊆ ( 𝑦 ) )
79 69 78 eqssd ( ( 𝑦 ∈ ( 𝐴𝑚 ) ∧ ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) → ( 𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
80 79 ralimiaa ( ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) ∈ ( 𝐹𝑦 ) → ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
81 54 56 80 3syl ( ( ( ( ( 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 ( 𝑦 ∈ ( 𝐴𝑚 ) → ¬ 𝑦𝑚 )
83 82 ad2antlr ( ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ¬ 𝑦𝑚 )
84 inss2 ( 𝐴𝑚 ) ⊆ 𝑚
85 simpr ( ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → 𝑛 ∈ ( 𝐴𝑚 ) )
86 84 85 sselid ( ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → 𝑛𝑚 )
87 eleq1 ( 𝑦 = 𝑛 → ( 𝑦𝑚𝑛𝑚 ) )
88 86 87 syl5ibrcom ( ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ( 𝑦 = 𝑛𝑦𝑚 ) )
89 83 88 mtod ( ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ¬ 𝑦 = 𝑛 )
90 89 58 syl ( ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) = ( 𝐹𝑦 ) )
91 90 iineq2dv ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴𝑚 ) ) → 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) = 𝑛 ∈ ( 𝐴𝑚 ) ( 𝐹𝑦 ) )
92 iinconst ( ( 𝐴𝑚 ) ≠ ∅ → 𝑛 ∈ ( 𝐴𝑚 ) ( 𝐹𝑦 ) = ( 𝐹𝑦 ) )
93 92 adantr ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴𝑚 ) ) → 𝑛 ∈ ( 𝐴𝑚 ) ( 𝐹𝑦 ) = ( 𝐹𝑦 ) )
94 91 93 eqtr2d ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴𝑚 ) ) → ( 𝐹𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
95 eqeq1 ( ( 𝑦 ) = ( 𝐹𝑦 ) → ( ( 𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ↔ ( 𝐹𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ) )
96 94 95 syl5ibrcom ( ( ( 𝐴𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴𝑚 ) ) → ( ( 𝑦 ) = ( 𝐹𝑦 ) → ( 𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ) )
97 96 ralimdva ( ( 𝐴𝑚 ) ≠ ∅ → ( ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) → ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ) )
98 5 97 mpan9 ( ( ( ( ( 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 ( ( 𝐴𝑚 ) ∪ ( 𝐴𝑚 ) ) = 𝐴
100 99 raleqi ( ∀ 𝑦 ∈ ( ( 𝐴𝑚 ) ∪ ( 𝐴𝑚 ) ) ( 𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) ↔ ∀ 𝑦𝐴 ( 𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
101 ralunb ( ∀ 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 ) ) ) )
102 100 101 bitr3i ( ∀ 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 ) ) ) )
103 81 98 102 sylanbrc ( ( ( ( ( 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 ( ∀ 𝑦𝐴 ( 𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) → X 𝑦𝐴 ( 𝑦 ) = X 𝑦𝐴 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
105 103 104 syl ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) → X 𝑦𝐴 ( 𝑦 ) = X 𝑦𝐴 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
106 ixpiin ( ( 𝐴𝑚 ) ≠ ∅ → X 𝑦𝐴 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) = 𝑛 ∈ ( 𝐴𝑚 ) X 𝑦𝐴 if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
107 106 adantl ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) → X 𝑦𝐴 𝑛 ∈ ( 𝐴𝑚 ) if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) = 𝑛 ∈ ( 𝐴𝑚 ) X 𝑦𝐴 if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
108 105 107 eqtrd ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) → X 𝑦𝐴 ( 𝑦 ) = 𝑛 ∈ ( 𝐴𝑚 ) X 𝑦𝐴 if ( 𝑦 = 𝑛 , ( 𝑛 ) , ( 𝐹𝑦 ) ) )
109 39 53 108 3eqtr4rd ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ ( 𝐴𝑚 ) ≠ ∅ ) → X 𝑦𝐴 ( 𝑦 ) = ( 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ) )
110 25 109 pm2.61dane ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) → X 𝑦𝐴 ( 𝑦 ) = ( 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ) )
111 ixpexg ( ∀ 𝑛𝐴 ( 𝐹𝑛 ) ∈ V → X 𝑛𝐴 ( 𝐹𝑛 ) ∈ V )
112 fvex ( 𝐹𝑛 ) ∈ V
113 112 uniex ( 𝐹𝑛 ) ∈ V
114 113 a1i ( 𝑛𝐴 ( 𝐹𝑛 ) ∈ V )
115 111 114 mprg X 𝑛𝐴 ( 𝐹𝑛 ) ∈ V
116 2 115 eqeltri 𝑋 ∈ V
117 116 mptex ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) ∈ V
118 117 cnvex ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) ∈ V
119 118 imaex ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ∈ V
120 119 dfiin2 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) = { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴𝑚 ) 𝑧 = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) }
121 inteq ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴𝑚 ) 𝑧 = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) } = ∅ → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴𝑚 ) 𝑧 = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) } = ∅ )
122 120 121 syl5eq ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴𝑚 ) 𝑧 = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) } = ∅ → 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) = ∅ )
123 int0 ∅ = V
124 122 123 eqtrdi ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴𝑚 ) 𝑧 = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) } = ∅ → 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) = V )
125 124 ineq2d ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴𝑚 ) 𝑧 = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) } = ∅ → ( 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ) = ( 𝑋 ∩ V ) )
126 inv1 ( 𝑋 ∩ V ) = 𝑋
127 125 126 eqtrdi ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴𝑚 ) 𝑧 = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) } = ∅ → ( 𝑋 𝑛 ∈ ( 𝐴𝑚 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ) = 𝑋 )
128 127 adantl ( ( ( ( ( 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 { 𝑋 } ∈ V
130 1 ptbas ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → 𝐵 ∈ TopBases )
131 1 2 ptpjpre2 ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝑘𝐴𝑢 ∈ ( 𝐹𝑘 ) ) ) → ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ∈ 𝐵 )
132 131 ralrimivva ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ∀ 𝑘𝐴𝑢 ∈ ( 𝐹𝑘 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ∈ 𝐵 )
133 eqid ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) = ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) )
134 133 fmpox ( ∀ 𝑘𝐴𝑢 ∈ ( 𝐹𝑘 ) ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ∈ 𝐵 ↔ ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) : 𝑘𝐴 ( { 𝑘 } × ( 𝐹𝑘 ) ) ⟶ 𝐵 )
135 132 134 sylib ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) : 𝑘𝐴 ( { 𝑘 } × ( 𝐹𝑘 ) ) ⟶ 𝐵 )
136 135 frnd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ⊆ 𝐵 )
137 130 136 ssexd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ∈ V )
138 unexg ( ( { 𝑋 } ∈ V ∧ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ∈ V ) → ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ∈ V )
139 129 137 138 sylancr ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ∈ V )
140 ssfii ( ( { 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 ) ) ) ) )
141 139 140 syl ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) )
142 141 ad2antrr ( ( ( ( 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 { 𝑋 } ⊆ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) )
144 116 snss ( 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ↔ { 𝑋 } ⊆ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) )
145 143 144 mpbir 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) )
146 145 a1i ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) → 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) )
147 142 146 sseldd ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) → 𝑋 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) )
148 147 adantr ( ( ( ( ( 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 ) ) ) ) )
149 128 148 eqeltrd ( ( ( ( ( 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 ) ) ) ) )
150 139 ad3antrrr ( ( ( ( ( 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 𝑛 ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) )
152 nfcv 𝑛 𝐴
153 nfcv 𝑛 ( 𝐹𝑘 )
154 nfixp1 𝑛 X 𝑛𝐴 ( 𝐹𝑛 )
155 2 154 nfcxfr 𝑛 𝑋
156 nfcv 𝑛 ( 𝑤𝑘 )
157 155 156 nfmpt 𝑛 ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) )
158 157 nfcnv 𝑛 ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) )
159 nfcv 𝑛 𝑢
160 158 159 nfima 𝑛 ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 )
161 152 153 160 nfmpo 𝑛 ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) )
162 161 nfrn 𝑛 ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) )
163 162 nfcri 𝑛 𝑧 ∈ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) )
164 df-ov ( 𝑛 ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ( 𝑛 ) ) = ( ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ‘ ⟨ 𝑛 , ( 𝑛 ) ⟩ )
165 119 a1i ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ∈ V )
166 fveq2 ( 𝑘 = 𝑛 → ( 𝑤𝑘 ) = ( 𝑤𝑛 ) )
167 166 mpteq2dv ( 𝑘 = 𝑛 → ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) = ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) )
168 167 cnveqd ( 𝑘 = 𝑛 ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) = ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) )
169 168 imaeq1d ( 𝑘 = 𝑛 → ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ 𝑢 ) )
170 imaeq2 ( 𝑢 = ( 𝑛 ) → ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ 𝑢 ) = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) )
171 169 170 sylan9eq ( ( 𝑘 = 𝑛𝑢 = ( 𝑛 ) ) → ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) )
172 fveq2 ( 𝑘 = 𝑛 → ( 𝐹𝑘 ) = ( 𝐹𝑛 ) )
173 171 172 133 ovmpox ( ( 𝑛𝐴 ∧ ( 𝑛 ) ∈ ( 𝐹𝑛 ) ∧ ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ∈ V ) → ( 𝑛 ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ( 𝑛 ) ) = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) )
174 29 35 165 173 syl3anc ( ( ( ( ( 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 ) ) )
175 164 174 eqtr3id ( ( ( ( ( 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 ) ) )
176 135 ad3antrrr ( ( ( ( ( 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 )
177 176 ffnd ( ( ( ( ( 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 ( ⟨ 𝑛 , ( 𝑛 ) ⟩ ∈ 𝑛𝐴 ( { 𝑛 } × ( 𝐹𝑛 ) ) ↔ ( 𝑛𝐴 ∧ ( 𝑛 ) ∈ ( 𝐹𝑛 ) ) )
179 29 35 178 sylanbrc ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ⟨ 𝑛 , ( 𝑛 ) ⟩ ∈ 𝑛𝐴 ( { 𝑛 } × ( 𝐹𝑛 ) ) )
180 sneq ( 𝑛 = 𝑘 → { 𝑛 } = { 𝑘 } )
181 fveq2 ( 𝑛 = 𝑘 → ( 𝐹𝑛 ) = ( 𝐹𝑘 ) )
182 180 181 xpeq12d ( 𝑛 = 𝑘 → ( { 𝑛 } × ( 𝐹𝑛 ) ) = ( { 𝑘 } × ( 𝐹𝑘 ) ) )
183 182 cbviunv 𝑛𝐴 ( { 𝑛 } × ( 𝐹𝑛 ) ) = 𝑘𝐴 ( { 𝑘 } × ( 𝐹𝑘 ) )
184 179 183 eleqtrdi ( ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴𝑚 ) ) → ⟨ 𝑛 , ( 𝑛 ) ⟩ ∈ 𝑘𝐴 ( { 𝑘 } × ( 𝐹𝑘 ) ) )
185 fnfvelrn ( ( ( 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 ) ) )
186 177 184 185 syl2anc ( ( ( ( ( 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 ) ) )
187 175 186 eqeltrrd ( ( ( ( ( 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 ( 𝑧 = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) → ( 𝑧 ∈ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ↔ ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) ∈ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) )
189 187 188 syl5ibrcom ( ( ( ( ( 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 ) ) ) )
190 189 ex ( ( ( ( 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 ) ) ) ) )
191 151 163 190 rexlimd ( ( ( ( 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 ) ) ) )
192 191 abssdv ( ( ( ( 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 ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ⊆ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) )
194 192 193 sstrdi ( ( ( ( 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 ) ) ) )
195 194 adantr ( ( ( ( ( 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 ( ( ( ( ( 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 ( ( ( ( ( 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 ( ( 𝑚 ∈ Fin ∧ ( 𝐴𝑚 ) ⊆ 𝑚 ) → ( 𝐴𝑚 ) ∈ Fin )
199 197 84 198 sylancl ( ( ( ( ( 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 ( ( 𝐴𝑚 ) ∈ Fin → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴𝑚 ) 𝑧 = ( ( 𝑤𝑋 ↦ ( 𝑤𝑛 ) ) “ ( 𝑛 ) ) } ∈ Fin )
201 199 200 syl ( ( ( ( ( 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 ( ( ( { 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 ) ) ) ) )
203 150 195 196 201 202 syl13anc ( ( ( ( ( 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 ) ) ) ) )
204 120 203 eqeltrid ( ( ( ( ( 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 ( 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 ) ) ) ) )
206 204 205 syl ( ( ( ( ( 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 ( ( { 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 ) ) ) ) )
208 139 207 syl ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) = ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) )
209 116 pwid 𝑋 ∈ 𝒫 𝑋
210 209 a1i ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → 𝑋 ∈ 𝒫 𝑋 )
211 210 snssd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → { 𝑋 } ⊆ 𝒫 𝑋 )
212 1 ptuni2 ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → X 𝑛𝐴 ( 𝐹𝑛 ) = 𝐵 )
213 2 212 syl5eq ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → 𝑋 = 𝐵 )
214 eqimss2 ( 𝑋 = 𝐵 𝐵𝑋 )
215 213 214 syl ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → 𝐵𝑋 )
216 sspwuni ( 𝐵 ⊆ 𝒫 𝑋 𝐵𝑋 )
217 215 216 sylibr ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → 𝐵 ⊆ 𝒫 𝑋 )
218 136 217 sstrd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ⊆ 𝒫 𝑋 )
219 211 218 unssd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝒫 𝑋 )
220 sspwuni ( ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝒫 𝑋 ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝑋 )
221 219 220 sylib ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝑋 )
222 elssuni ( 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) → 𝑋 ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) )
223 145 222 mp1i ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → 𝑋 ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) )
224 221 223 eqssd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) = 𝑋 )
225 208 224 eqtr3d ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) = 𝑋 )
226 225 ad3antrrr ( ( ( ( ( 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 )
227 206 226 sseqtrd ( ( ( ( ( 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 )
228 227 52 sylib ( ( ( ( ( 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 ) ) )
229 228 204 eqeltrd ( ( ( ( ( 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 ) ) ) ) )
230 149 229 pm2.61dane ( ( ( ( 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 ) ) ) ) )
231 110 230 eqeltrd ( ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) → X 𝑦𝐴 ( 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) )
232 231 rexlimdvaa ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ) → ( ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) → X 𝑦𝐴 ( 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) ) )
233 232 impr ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) → X 𝑦𝐴 ( 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) )
234 4 233 sylan2b ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) → X 𝑦𝐴 ( 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) )
235 eleq1 ( 𝑠 = X 𝑦𝐴 ( 𝑦 ) → ( 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) ↔ X 𝑦𝐴 ( 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) ) )
236 234 235 syl5ibrcom ( ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) ∧ ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ) → ( 𝑠 = X 𝑦𝐴 ( 𝑦 ) → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) ) )
237 236 expimpd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( ( ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ∧ 𝑠 = X 𝑦𝐴 ( 𝑦 ) ) → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) ) )
238 237 exlimdv ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( ∃ ( ( Fn 𝐴 ∧ ∀ 𝑦𝐴 ( 𝑦 ) ∈ ( 𝐹𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴𝑚 ) ( 𝑦 ) = ( 𝐹𝑦 ) ) ∧ 𝑠 = X 𝑦𝐴 ( 𝑦 ) ) → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) ) )
239 3 238 syl5bi ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( 𝑠𝐵𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) ) )
240 239 ssrdv ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → 𝐵 ⊆ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) )
241 1 ptbasid ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → X 𝑛𝐴 ( 𝐹𝑛 ) ∈ 𝐵 )
242 2 241 eqeltrid ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → 𝑋𝐵 )
243 242 snssd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → { 𝑋 } ⊆ 𝐵 )
244 243 136 unssd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝐵 )
245 fiss ( ( 𝐵 ∈ TopBases ∧ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝐵 ) → ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) ⊆ ( fi ‘ 𝐵 ) )
246 130 244 245 syl2anc ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) ⊆ ( fi ‘ 𝐵 ) )
247 1 ptbasin2 ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ 𝐵 ) = 𝐵 )
248 246 247 sseqtrd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) ⊆ 𝐵 )
249 240 248 eqssd ( ( 𝐴𝑉𝐹 : 𝐴 ⟶ Top ) → 𝐵 = ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘𝐴 , 𝑢 ∈ ( 𝐹𝑘 ) ↦ ( ( 𝑤𝑋 ↦ ( 𝑤𝑘 ) ) “ 𝑢 ) ) ) ) )