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 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) }
ptbasfi.2 ⊢ 𝑋 = X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 )
Assertion ptbasfi ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐵 = ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 ptbas.1 ⊢ 𝐵 = { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) }
2 ptbasfi.2 ⊢ 𝑋 = X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 )
3 1 elpt ⊢ ( 𝑠 ∈ 𝐵 ↔ ∃ ℎ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑠 = X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ) )
4 df-3an ⊢ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) )
5 simprr ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) )
6 disjdif2 ⊢ ( ( 𝐴 ∩ 𝑚 ) = ∅ → ( 𝐴 ∖ 𝑚 ) = 𝐴 )
7 6 raleqdv ⊢ ( ( 𝐴 ∩ 𝑚 ) = ∅ → ( ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) )
8 7 biimpac ⊢ ( ( ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) )
9 ixpeq2 ⊢ ( ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) )
10 8 9 syl ⊢ ( ( ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) )
11 fveq2 ⊢ ( 𝑛 = 𝑦 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑦 ) )
12 11 unieqd ⊢ ( 𝑛 = 𝑦 → ∪ ( 𝐹 ‘ 𝑛 ) = ∪ ( 𝐹 ‘ 𝑦 ) )
13 12 cbvixpv ⊢ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 )
14 2 13 eqtri ⊢ 𝑋 = X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 )
15 10 14 eqtr4di ⊢ ( ( ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = 𝑋 )
16 5 15 sylan ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = 𝑋 )
17 ssv ⊢ 𝑋 ⊆ V
18 iineq1 ⊢ ( ( 𝐴 ∩ 𝑚 ) = ∅ → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = ∩ 𝑛 ∈ ∅ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
19 0iin ⊢ ∩ 𝑛 ∈ ∅ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = V
20 18 19 eqtrdi ⊢ ( ( 𝐴 ∩ 𝑚 ) = ∅ → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = V )
21 17 20 sseqtrrid ⊢ ( ( 𝐴 ∩ 𝑚 ) = ∅ → 𝑋 ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
22 21 adantl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → 𝑋 ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
23 df-ss ⊢ ( 𝑋 ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ↔ ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = 𝑋 )
24 22 23 sylib ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = 𝑋 )
25 16 24 eqtr4d ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) )
26 simplll ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) )
27 inss1 ⊢ ( 𝐴 ∩ 𝑚 ) ⊆ 𝐴
28 simpr ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) )
29 27 28 sseldi ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → 𝑛 ∈ 𝐴 )
30 fveq2 ⊢ ( 𝑦 = 𝑛 → ( ℎ ‘ 𝑦 ) = ( ℎ ‘ 𝑛 ) )
31 fveq2 ⊢ ( 𝑦 = 𝑛 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑛 ) )
32 30 31 eleq12d ⊢ ( 𝑦 = 𝑛 → ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ↔ ( ℎ ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) ) )
33 simprr ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) → ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) )
34 33 ad2antrr ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) )
35 32 34 29 rspcdva ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ℎ ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) )
36 14 ptpjpre1 ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝑛 ∈ 𝐴 ∧ ( ℎ ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
37 26 29 35 36 syl12anc ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
38 37 adantlr ⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
39 38 iineq2dv ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
40 simpr ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ( 𝐴 ∩ 𝑚 ) ≠ ∅ )
41 cnvimass ⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ dom ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) )
42 eqid ⊢ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) = ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) )
43 42 dmmptss ⊢ dom ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) ⊆ 𝑋
44 41 43 sstri ⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ 𝑋
45 44 14 sseqtri ⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 )
46 45 rgenw ⊢ ∀ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 )
47 r19.2z ⊢ ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) ) → ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) )
48 40 46 47 sylancl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) )
49 iinss ⊢ ( ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) )
50 48 49 syl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) )
51 50 14 sseqtrrdi ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ 𝑋 )
52 sseqin2 ⊢ ( ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ 𝑋 ↔ ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
53 51 52 sylib ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
54 33 ad2antrr ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) )
55 ssralv ⊢ ( ( 𝐴 ∩ 𝑚 ) ⊆ 𝐴 → ( ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) )
56 27 55 ax-mp ⊢ ( ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) )
57 elssuni ⊢ ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ( ℎ ‘ 𝑦 ) ⊆ ∪ ( 𝐹 ‘ 𝑦 ) )
58 iffalse ⊢ ( ¬ 𝑦 = 𝑛 → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ∪ ( 𝐹 ‘ 𝑦 ) )
59 58 sseq2d ⊢ ( ¬ 𝑦 = 𝑛 → ( ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ( ℎ ‘ 𝑦 ) ⊆ ∪ ( 𝐹 ‘ 𝑦 ) ) )
60 57 59 syl5ibrcom ⊢ ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ( ¬ 𝑦 = 𝑛 → ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) )
61 ssid ⊢ ( ℎ ‘ 𝑦 ) ⊆ ( ℎ ‘ 𝑦 )
62 iftrue ⊢ ( 𝑦 = 𝑛 → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ( ℎ ‘ 𝑛 ) )
63 62 30 eqtr4d ⊢ ( 𝑦 = 𝑛 → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ( ℎ ‘ 𝑦 ) )
64 61 63 sseqtrrid ⊢ ( 𝑦 = 𝑛 → ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
65 60 64 pm2.61d2 ⊢ ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
66 65 ralrimivw ⊢ ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ∀ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
67 ssiin ⊢ ( ( ℎ ‘ 𝑦 ) ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
68 66 67 sylibr ⊢ ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ( ℎ ‘ 𝑦 ) ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
69 68 adantl ⊢ ( ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ∧ ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) → ( ℎ ‘ 𝑦 ) ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
70 62 equcoms ⊢ ( 𝑛 = 𝑦 → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ( ℎ ‘ 𝑛 ) )
71 fveq2 ⊢ ( 𝑛 = 𝑦 → ( ℎ ‘ 𝑛 ) = ( ℎ ‘ 𝑦 ) )
72 70 71 eqtrd ⊢ ( 𝑛 = 𝑦 → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ( ℎ ‘ 𝑦 ) )
73 72 sseq1d ⊢ ( 𝑛 = 𝑦 → ( if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) ↔ ( ℎ ‘ 𝑦 ) ⊆ ( ℎ ‘ 𝑦 ) ) )
74 73 rspcev ⊢ ( ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ∧ ( ℎ ‘ 𝑦 ) ⊆ ( ℎ ‘ 𝑦 ) ) → ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) )
75 61 74 mpan2 ⊢ ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) → ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) )
76 iinss ⊢ ( ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) )
77 75 76 syl ⊢ ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) )
78 77 adantr ⊢ ( ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ∧ ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) )
79 69 78 eqssd ⊢ ( ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ∧ ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) → ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
80 79 ralimiaa ⊢ ( ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
81 54 56 80 3syl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
82 eldifn ⊢ ( 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) → ¬ 𝑦 ∈ 𝑚 )
83 82 ad2antlr ⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ¬ 𝑦 ∈ 𝑚 )
84 inss2 ⊢ ( 𝐴 ∩ 𝑚 ) ⊆ 𝑚
85 simpr ⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) )
86 84 85 sseldi ⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → 𝑛 ∈ 𝑚 )
87 eleq1 ⊢ ( 𝑦 = 𝑛 → ( 𝑦 ∈ 𝑚 ↔ 𝑛 ∈ 𝑚 ) )
88 86 87 syl5ibrcom ⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝑦 = 𝑛 → 𝑦 ∈ 𝑚 ) )
89 83 88 mtod ⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ¬ 𝑦 = 𝑛 )
90 89 58 syl ⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ∪ ( 𝐹 ‘ 𝑦 ) )
91 90 iineq2dv ⊢ ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐹 ‘ 𝑦 ) )
92 iinconst ⊢ ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐹 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) )
93 92 adantr ⊢ ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐹 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) )
94 91 93 eqtr2d ⊢ ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) → ∪ ( 𝐹 ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
95 eqeq1 ⊢ ( ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) → ( ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ∪ ( 𝐹 ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) )
96 94 95 syl5ibrcom ⊢ ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) → ( ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) → ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) )
97 96 ralimdva ⊢ ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ → ( ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) → ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) )
98 5 97 mpan9 ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
99 inundif ⊢ ( ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐴 ∖ 𝑚 ) ) = 𝐴
100 99 raleqi ⊢ ( ∀ 𝑦 ∈ ( ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐴 ∖ 𝑚 ) ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
101 ralunb ⊢ ( ∀ 𝑦 ∈ ( ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐴 ∖ 𝑚 ) ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) )
102 100 101 bitr3i ⊢ ( ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) )
103 81 98 102 sylanbrc ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
104 ixpeq2 ⊢ ( ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = X 𝑦 ∈ 𝐴 ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
105 103 104 syl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = X 𝑦 ∈ 𝐴 ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
106 ixpiin ⊢ ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ → X 𝑦 ∈ 𝐴 ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
107 106 adantl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → X 𝑦 ∈ 𝐴 ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
108 105 107 eqtrd ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) )
109 39 53 108 3eqtr4rd ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) )
110 25 109 pm2.61dane ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) )
111 ixpexg ⊢ ( ∀ 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ∈ V → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ∈ V )
112 fvex ⊢ ( 𝐹 ‘ 𝑛 ) ∈ V
113 112 uniex ⊢ ∪ ( 𝐹 ‘ 𝑛 ) ∈ V
114 113 a1i ⊢ ( 𝑛 ∈ 𝐴 → ∪ ( 𝐹 ‘ 𝑛 ) ∈ V )
115 111 114 mprg ⊢ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ∈ V
116 2 115 eqeltri ⊢ 𝑋 ∈ V
117 116 mptex ⊢ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) ∈ V
118 117 cnvex ⊢ ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) ∈ V
119 118 imaex ⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ V
120 119 dfiin2 ⊢ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = ∩ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) }
121 inteq ⊢ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ → ∩ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∩ ∅ )
122 120 121 syl5eq ⊢ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = ∩ ∅ )
123 int0 ⊢ ∩ ∅ = V
124 122 123 eqtrdi ⊢ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = V )
125 124 ineq2d ⊢ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = ( 𝑋 ∩ V ) )
126 inv1 ⊢ ( 𝑋 ∩ V ) = 𝑋
127 125 126 eqtrdi ⊢ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = 𝑋 )
128 127 adantl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = 𝑋 )
129 snex ⊢ { 𝑋 } ∈ V
130 1 ptbas ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐵 ∈ TopBases )
131 1 2 ptpjpre2 ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝐵 )
132 131 ralrimivva ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∀ 𝑘 ∈ 𝐴 ∀ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝐵 )
133 eqid ⊢ ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) = ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) )
134 133 fmpox ⊢ ( ∀ 𝑘 ∈ 𝐴 ∀ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝐵 ↔ ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) : ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ⟶ 𝐵 )
135 132 134 sylib ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) : ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ⟶ 𝐵 )
136 135 frnd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ⊆ 𝐵 )
137 130 136 ssexd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ∈ V )
138 unexg ⊢ ( ( { 𝑋 } ∈ V ∧ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ∈ V ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V )
139 129 137 138 sylancr ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V )
140 ssfii ⊢ ( ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
141 139 140 syl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
142 141 ad2antrr ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
143 ssun1 ⊢ { 𝑋 } ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) )
144 116 snss ⊢ ( 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ↔ { 𝑋 } ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) )
145 143 144 mpbir ⊢ 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) )
146 145 a1i ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) )
147 142 146 sseldd ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → 𝑋 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
148 147 adantr ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ ) → 𝑋 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
149 128 148 eqeltrd ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
150 139 ad3antrrr ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V )
151 nfv ⊢ Ⅎ 𝑛 ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) )
152 nfcv ⊢ Ⅎ 𝑛 𝐴
153 nfcv ⊢ Ⅎ 𝑛 ( 𝐹 ‘ 𝑘 )
154 nfixp1 ⊢ Ⅎ 𝑛 X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 )
155 2 154 nfcxfr ⊢ Ⅎ 𝑛 𝑋
156 nfcv ⊢ Ⅎ 𝑛 ( 𝑤 ‘ 𝑘 )
157 155 156 nfmpt ⊢ Ⅎ 𝑛 ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) )
158 157 nfcnv ⊢ Ⅎ 𝑛 ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) )
159 nfcv ⊢ Ⅎ 𝑛 𝑢
160 158 159 nfima ⊢ Ⅎ 𝑛 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 )
161 152 153 160 nfmpo ⊢ Ⅎ 𝑛 ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) )
162 161 nfrn ⊢ Ⅎ 𝑛 ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) )
163 162 nfcri ⊢ Ⅎ 𝑛 𝑧 ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) )
164 df-ov ⊢ ( 𝑛 ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ( ℎ ‘ 𝑛 ) ) = ( ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ‘ ⟨ 𝑛 , ( ℎ ‘ 𝑛 ) ⟩ )
165 119 a1i ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ V )
166 fveq2 ⊢ ( 𝑘 = 𝑛 → ( 𝑤 ‘ 𝑘 ) = ( 𝑤 ‘ 𝑛 ) )
167 166 mpteq2dv ⊢ ( 𝑘 = 𝑛 → ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) = ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) )
168 167 cnveqd ⊢ ( 𝑘 = 𝑛 → ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) = ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) )
169 168 imaeq1d ⊢ ( 𝑘 = 𝑛 → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ 𝑢 ) )
170 imaeq2 ⊢ ( 𝑢 = ( ℎ ‘ 𝑛 ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ 𝑢 ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
171 169 170 sylan9eq ⊢ ( ( 𝑘 = 𝑛 ∧ 𝑢 = ( ℎ ‘ 𝑛 ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
172 fveq2 ⊢ ( 𝑘 = 𝑛 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑛 ) )
173 171 172 133 ovmpox ⊢ ( ( 𝑛 ∈ 𝐴 ∧ ( ℎ ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) ∧ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ V ) → ( 𝑛 ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ( ℎ ‘ 𝑛 ) ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
174 29 35 165 173 syl3anc ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝑛 ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ( ℎ ‘ 𝑛 ) ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
175 164 174 eqtr3id ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ‘ ⟨ 𝑛 , ( ℎ ‘ 𝑛 ) ⟩ ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
176 135 ad3antrrr ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) : ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ⟶ 𝐵 )
177 176 ffnd ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) Fn ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) )
178 opeliunxp ⊢ ( ⟨ 𝑛 , ( ℎ ‘ 𝑛 ) ⟩ ∈ ∪ 𝑛 ∈ 𝐴 ( { 𝑛 } × ( 𝐹 ‘ 𝑛 ) ) ↔ ( 𝑛 ∈ 𝐴 ∧ ( ℎ ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) ) )
179 29 35 178 sylanbrc ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ⟨ 𝑛 , ( ℎ ‘ 𝑛 ) ⟩ ∈ ∪ 𝑛 ∈ 𝐴 ( { 𝑛 } × ( 𝐹 ‘ 𝑛 ) ) )
180 sneq ⊢ ( 𝑛 = 𝑘 → { 𝑛 } = { 𝑘 } )
181 fveq2 ⊢ ( 𝑛 = 𝑘 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑘 ) )
182 180 181 xpeq12d ⊢ ( 𝑛 = 𝑘 → ( { 𝑛 } × ( 𝐹 ‘ 𝑛 ) ) = ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) )
183 182 cbviunv ⊢ ∪ 𝑛 ∈ 𝐴 ( { 𝑛 } × ( 𝐹 ‘ 𝑛 ) ) = ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) )
184 179 183 eleqtrdi ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ⟨ 𝑛 , ( ℎ ‘ 𝑛 ) ⟩ ∈ ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) )
185 fnfvelrn ⊢ ( ( ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) Fn ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ∧ ⟨ 𝑛 , ( ℎ ‘ 𝑛 ) ⟩ ∈ ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ‘ ⟨ 𝑛 , ( ℎ ‘ 𝑛 ) ⟩ ) ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) )
186 177 184 185 syl2anc ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ‘ ⟨ 𝑛 , ( ℎ ‘ 𝑛 ) ⟩ ) ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) )
187 175 186 eqeltrrd ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) )
188 eleq1 ⊢ ( 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) → ( 𝑧 ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ↔ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) )
189 187 188 syl5ibrcom ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) → 𝑧 ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) )
190 189 ex ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ( 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) → ( 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) → 𝑧 ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
191 151 163 190 rexlimd ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ( ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) → 𝑧 ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) )
192 191 abssdv ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ⊆ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) )
193 ssun2 ⊢ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) )
194 192 193 sstrdi ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) )
195 194 adantr ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) )
196 simpr ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ )
197 simplrl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → 𝑚 ∈ Fin )
198 ssfi ⊢ ( ( 𝑚 ∈ Fin ∧ ( 𝐴 ∩ 𝑚 ) ⊆ 𝑚 ) → ( 𝐴 ∩ 𝑚 ) ∈ Fin )
199 197 84 198 sylancl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ( 𝐴 ∩ 𝑚 ) ∈ Fin )
200 abrexfi ⊢ ( ( 𝐴 ∩ 𝑚 ) ∈ Fin → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ∈ Fin )
201 199 200 syl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ∈ Fin )
202 elfir ⊢ ( ( ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V ∧ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ∈ Fin ) ) → ∩ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
203 150 195 196 201 202 syl13anc ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ∩ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
204 120 203 eqeltrid ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
205 elssuni ⊢ ( ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
206 204 205 syl ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
207 fiuni ⊢ ( ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V → ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) = ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
208 139 207 syl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) = ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
209 116 pwid ⊢ 𝑋 ∈ 𝒫 𝑋
210 209 a1i ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝑋 ∈ 𝒫 𝑋 )
211 210 snssd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → { 𝑋 } ⊆ 𝒫 𝑋 )
212 1 ptuni2 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = ∪ 𝐵 )
213 2 212 syl5eq ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝑋 = ∪ 𝐵 )
214 eqimss2 ⊢ ( 𝑋 = ∪ 𝐵 → ∪ 𝐵 ⊆ 𝑋 )
215 213 214 syl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ 𝐵 ⊆ 𝑋 )
216 sspwuni ⊢ ( 𝐵 ⊆ 𝒫 𝑋 ↔ ∪ 𝐵 ⊆ 𝑋 )
217 215 216 sylibr ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐵 ⊆ 𝒫 𝑋 )
218 136 217 sstrd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ⊆ 𝒫 𝑋 )
219 211 218 unssd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝒫 𝑋 )
220 sspwuni ⊢ ( ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝒫 𝑋 ↔ ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝑋 )
221 219 220 sylib ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝑋 )
222 elssuni ⊢ ( 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) → 𝑋 ⊆ ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) )
223 145 222 mp1i ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝑋 ⊆ ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) )
224 221 223 eqssd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) = 𝑋 )
225 208 224 eqtr3d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) = 𝑋 )
226 225 ad3antrrr ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) = 𝑋 )
227 206 226 sseqtrd ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ 𝑋 )
228 227 52 sylib ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) )
229 228 204 eqeltrd ⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
230 149 229 pm2.61dane ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
231 110 230 eqeltrd ⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
232 231 rexlimdvaa ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) → ( ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) )
233 232 impr ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
234 4 233 sylan2b ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
235 eleq1 ⊢ ( 𝑠 = X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) → ( 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ↔ X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) )
236 234 235 syl5ibrcom ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ( 𝑠 = X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) )
237 236 expimpd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑠 = X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ) → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) )
238 237 exlimdv ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( ∃ ℎ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑠 = X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ) → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) )
239 3 238 syl5bi ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( 𝑠 ∈ 𝐵 → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) )
240 239 ssrdv ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐵 ⊆ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
241 1 ptbasid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ∈ 𝐵 )
242 2 241 eqeltrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝑋 ∈ 𝐵 )
243 242 snssd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → { 𝑋 } ⊆ 𝐵 )
244 243 136 unssd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝐵 )
245 fiss ⊢ ( ( 𝐵 ∈ TopBases ∧ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝐵 ) → ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ⊆ ( fi ‘ 𝐵 ) )
246 130 244 245 syl2anc ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ⊆ ( fi ‘ 𝐵 ) )
247 1 ptbasin2 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ 𝐵 ) = 𝐵 )
248 246 247 sseqtrd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ⊆ 𝐵 )
249 240 248 eqssd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐵 = ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )