Metamath Proof Explorer


Theorem wuncval2

Description: Our earlier expression for a containing weak universe is in fact the weak universe closure. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses wuncval2.f 𝐹 = ( rec ( ( 𝑧 ∈ V ↦ ( ( 𝑧 𝑧 ) ∪ 𝑥𝑧 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) ) ) ) , ( 𝐴 ∪ 1o ) ) ↾ ω )
wuncval2.u 𝑈 = ran 𝐹
Assertion wuncval2 ( 𝐴𝑉 → ( wUniCl ‘ 𝐴 ) = 𝑈 )

Proof

Step Hyp Ref Expression
1 wuncval2.f 𝐹 = ( rec ( ( 𝑧 ∈ V ↦ ( ( 𝑧 𝑧 ) ∪ 𝑥𝑧 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) ) ) ) , ( 𝐴 ∪ 1o ) ) ↾ ω )
2 wuncval2.u 𝑈 = ran 𝐹
3 1 2 wunex2 ( 𝐴𝑉 → ( 𝑈 ∈ WUni ∧ 𝐴𝑈 ) )
4 wuncss ( ( 𝑈 ∈ WUni ∧ 𝐴𝑈 ) → ( wUniCl ‘ 𝐴 ) ⊆ 𝑈 )
5 3 4 syl ( 𝐴𝑉 → ( wUniCl ‘ 𝐴 ) ⊆ 𝑈 )
6 frfnom ( rec ( ( 𝑧 ∈ V ↦ ( ( 𝑧 𝑧 ) ∪ 𝑥𝑧 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) ) ) ) , ( 𝐴 ∪ 1o ) ) ↾ ω ) Fn ω
7 1 fneq1i ( 𝐹 Fn ω ↔ ( rec ( ( 𝑧 ∈ V ↦ ( ( 𝑧 𝑧 ) ∪ 𝑥𝑧 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) ) ) ) , ( 𝐴 ∪ 1o ) ) ↾ ω ) Fn ω )
8 6 7 mpbir 𝐹 Fn ω
9 fniunfv ( 𝐹 Fn ω → 𝑚 ∈ ω ( 𝐹𝑚 ) = ran 𝐹 )
10 8 9 ax-mp 𝑚 ∈ ω ( 𝐹𝑚 ) = ran 𝐹
11 2 10 eqtr4i 𝑈 = 𝑚 ∈ ω ( 𝐹𝑚 )
12 fveq2 ( 𝑚 = ∅ → ( 𝐹𝑚 ) = ( 𝐹 ‘ ∅ ) )
13 12 sseq1d ( 𝑚 = ∅ → ( ( 𝐹𝑚 ) ⊆ ( wUniCl ‘ 𝐴 ) ↔ ( 𝐹 ‘ ∅ ) ⊆ ( wUniCl ‘ 𝐴 ) ) )
14 fveq2 ( 𝑚 = 𝑛 → ( 𝐹𝑚 ) = ( 𝐹𝑛 ) )
15 14 sseq1d ( 𝑚 = 𝑛 → ( ( 𝐹𝑚 ) ⊆ ( wUniCl ‘ 𝐴 ) ↔ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) )
16 fveq2 ( 𝑚 = suc 𝑛 → ( 𝐹𝑚 ) = ( 𝐹 ‘ suc 𝑛 ) )
17 16 sseq1d ( 𝑚 = suc 𝑛 → ( ( 𝐹𝑚 ) ⊆ ( wUniCl ‘ 𝐴 ) ↔ ( 𝐹 ‘ suc 𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) )
18 1on 1o ∈ On
19 unexg ( ( 𝐴𝑉 ∧ 1o ∈ On ) → ( 𝐴 ∪ 1o ) ∈ V )
20 18 19 mpan2 ( 𝐴𝑉 → ( 𝐴 ∪ 1o ) ∈ V )
21 1 fveq1i ( 𝐹 ‘ ∅ ) = ( ( rec ( ( 𝑧 ∈ V ↦ ( ( 𝑧 𝑧 ) ∪ 𝑥𝑧 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) ) ) ) , ( 𝐴 ∪ 1o ) ) ↾ ω ) ‘ ∅ )
22 fr0g ( ( 𝐴 ∪ 1o ) ∈ V → ( ( rec ( ( 𝑧 ∈ V ↦ ( ( 𝑧 𝑧 ) ∪ 𝑥𝑧 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) ) ) ) , ( 𝐴 ∪ 1o ) ) ↾ ω ) ‘ ∅ ) = ( 𝐴 ∪ 1o ) )
23 21 22 syl5eq ( ( 𝐴 ∪ 1o ) ∈ V → ( 𝐹 ‘ ∅ ) = ( 𝐴 ∪ 1o ) )
24 20 23 syl ( 𝐴𝑉 → ( 𝐹 ‘ ∅ ) = ( 𝐴 ∪ 1o ) )
25 wuncid ( 𝐴𝑉𝐴 ⊆ ( wUniCl ‘ 𝐴 ) )
26 df1o2 1o = { ∅ }
27 wunccl ( 𝐴𝑉 → ( wUniCl ‘ 𝐴 ) ∈ WUni )
28 27 wun0 ( 𝐴𝑉 → ∅ ∈ ( wUniCl ‘ 𝐴 ) )
29 28 snssd ( 𝐴𝑉 → { ∅ } ⊆ ( wUniCl ‘ 𝐴 ) )
30 26 29 eqsstrid ( 𝐴𝑉 → 1o ⊆ ( wUniCl ‘ 𝐴 ) )
31 25 30 unssd ( 𝐴𝑉 → ( 𝐴 ∪ 1o ) ⊆ ( wUniCl ‘ 𝐴 ) )
32 24 31 eqsstrd ( 𝐴𝑉 → ( 𝐹 ‘ ∅ ) ⊆ ( wUniCl ‘ 𝐴 ) )
33 simplr ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) → 𝑛 ∈ ω )
34 fvex ( 𝐹𝑛 ) ∈ V
35 34 uniex ( 𝐹𝑛 ) ∈ V
36 34 35 unex ( ( 𝐹𝑛 ) ∪ ( 𝐹𝑛 ) ) ∈ V
37 prex { 𝒫 𝑢 , 𝑢 } ∈ V
38 34 mptex ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ∈ V
39 38 rnex ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ∈ V
40 37 39 unex ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ∈ V
41 34 40 iunex 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ∈ V
42 36 41 unex ( ( ( 𝐹𝑛 ) ∪ ( 𝐹𝑛 ) ) ∪ 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ) ∈ V
43 id ( 𝑤 = 𝑧𝑤 = 𝑧 )
44 unieq ( 𝑤 = 𝑧 𝑤 = 𝑧 )
45 43 44 uneq12d ( 𝑤 = 𝑧 → ( 𝑤 𝑤 ) = ( 𝑧 𝑧 ) )
46 pweq ( 𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥 )
47 unieq ( 𝑢 = 𝑥 𝑢 = 𝑥 )
48 46 47 preq12d ( 𝑢 = 𝑥 → { 𝒫 𝑢 , 𝑢 } = { 𝒫 𝑥 , 𝑥 } )
49 preq1 ( 𝑢 = 𝑥 → { 𝑢 , 𝑣 } = { 𝑥 , 𝑣 } )
50 49 mpteq2dv ( 𝑢 = 𝑥 → ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) = ( 𝑣𝑤 ↦ { 𝑥 , 𝑣 } ) )
51 50 rneqd ( 𝑢 = 𝑥 → ran ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) = ran ( 𝑣𝑤 ↦ { 𝑥 , 𝑣 } ) )
52 48 51 uneq12d ( 𝑢 = 𝑥 → ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) ) = ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑣𝑤 ↦ { 𝑥 , 𝑣 } ) ) )
53 52 cbviunv 𝑢𝑤 ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) ) = 𝑥𝑤 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑣𝑤 ↦ { 𝑥 , 𝑣 } ) )
54 preq2 ( 𝑣 = 𝑦 → { 𝑥 , 𝑣 } = { 𝑥 , 𝑦 } )
55 54 cbvmptv ( 𝑣𝑤 ↦ { 𝑥 , 𝑣 } ) = ( 𝑦𝑤 ↦ { 𝑥 , 𝑦 } )
56 mpteq1 ( 𝑤 = 𝑧 → ( 𝑦𝑤 ↦ { 𝑥 , 𝑦 } ) = ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) )
57 55 56 syl5eq ( 𝑤 = 𝑧 → ( 𝑣𝑤 ↦ { 𝑥 , 𝑣 } ) = ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) )
58 57 rneqd ( 𝑤 = 𝑧 → ran ( 𝑣𝑤 ↦ { 𝑥 , 𝑣 } ) = ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) )
59 58 uneq2d ( 𝑤 = 𝑧 → ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑣𝑤 ↦ { 𝑥 , 𝑣 } ) ) = ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) ) )
60 43 59 iuneq12d ( 𝑤 = 𝑧 𝑥𝑤 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑣𝑤 ↦ { 𝑥 , 𝑣 } ) ) = 𝑥𝑧 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) ) )
61 53 60 syl5eq ( 𝑤 = 𝑧 𝑢𝑤 ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) ) = 𝑥𝑧 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) ) )
62 45 61 uneq12d ( 𝑤 = 𝑧 → ( ( 𝑤 𝑤 ) ∪ 𝑢𝑤 ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) ) ) = ( ( 𝑧 𝑧 ) ∪ 𝑥𝑧 ( { 𝒫 𝑥 , 𝑥 } ∪ ran ( 𝑦𝑧 ↦ { 𝑥 , 𝑦 } ) ) ) )
63 id ( 𝑤 = ( 𝐹𝑛 ) → 𝑤 = ( 𝐹𝑛 ) )
64 unieq ( 𝑤 = ( 𝐹𝑛 ) → 𝑤 = ( 𝐹𝑛 ) )
65 63 64 uneq12d ( 𝑤 = ( 𝐹𝑛 ) → ( 𝑤 𝑤 ) = ( ( 𝐹𝑛 ) ∪ ( 𝐹𝑛 ) ) )
66 mpteq1 ( 𝑤 = ( 𝐹𝑛 ) → ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) = ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) )
67 66 rneqd ( 𝑤 = ( 𝐹𝑛 ) → ran ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) = ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) )
68 67 uneq2d ( 𝑤 = ( 𝐹𝑛 ) → ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) ) = ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) )
69 63 68 iuneq12d ( 𝑤 = ( 𝐹𝑛 ) → 𝑢𝑤 ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) ) = 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) )
70 65 69 uneq12d ( 𝑤 = ( 𝐹𝑛 ) → ( ( 𝑤 𝑤 ) ∪ 𝑢𝑤 ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣𝑤 ↦ { 𝑢 , 𝑣 } ) ) ) = ( ( ( 𝐹𝑛 ) ∪ ( 𝐹𝑛 ) ) ∪ 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ) )
71 1 62 70 frsucmpt2 ( ( 𝑛 ∈ ω ∧ ( ( ( 𝐹𝑛 ) ∪ ( 𝐹𝑛 ) ) ∪ 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ) ∈ V ) → ( 𝐹 ‘ suc 𝑛 ) = ( ( ( 𝐹𝑛 ) ∪ ( 𝐹𝑛 ) ) ∪ 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ) )
72 33 42 71 sylancl ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) → ( 𝐹 ‘ suc 𝑛 ) = ( ( ( 𝐹𝑛 ) ∪ ( 𝐹𝑛 ) ) ∪ 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ) )
73 simpr ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) → ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) )
74 27 ad3antrrr ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) → ( wUniCl ‘ 𝐴 ) ∈ WUni )
75 73 sselda ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) → 𝑢 ∈ ( wUniCl ‘ 𝐴 ) )
76 74 75 wunelss ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) → 𝑢 ⊆ ( wUniCl ‘ 𝐴 ) )
77 76 ralrimiva ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) → ∀ 𝑢 ∈ ( 𝐹𝑛 ) 𝑢 ⊆ ( wUniCl ‘ 𝐴 ) )
78 unissb ( ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ↔ ∀ 𝑢 ∈ ( 𝐹𝑛 ) 𝑢 ⊆ ( wUniCl ‘ 𝐴 ) )
79 77 78 sylibr ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) → ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) )
80 73 79 unssd ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) → ( ( 𝐹𝑛 ) ∪ ( 𝐹𝑛 ) ) ⊆ ( wUniCl ‘ 𝐴 ) )
81 74 75 wunpw ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) → 𝒫 𝑢 ∈ ( wUniCl ‘ 𝐴 ) )
82 74 75 wununi ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) → 𝑢 ∈ ( wUniCl ‘ 𝐴 ) )
83 81 82 prssd ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) → { 𝒫 𝑢 , 𝑢 } ⊆ ( wUniCl ‘ 𝐴 ) )
84 74 adantr ( ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) ∧ 𝑣 ∈ ( 𝐹𝑛 ) ) → ( wUniCl ‘ 𝐴 ) ∈ WUni )
85 75 adantr ( ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) ∧ 𝑣 ∈ ( 𝐹𝑛 ) ) → 𝑢 ∈ ( wUniCl ‘ 𝐴 ) )
86 simplr ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) → ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) )
87 86 sselda ( ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) ∧ 𝑣 ∈ ( 𝐹𝑛 ) ) → 𝑣 ∈ ( wUniCl ‘ 𝐴 ) )
88 84 85 87 wunpr ( ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) ∧ 𝑣 ∈ ( 𝐹𝑛 ) ) → { 𝑢 , 𝑣 } ∈ ( wUniCl ‘ 𝐴 ) )
89 88 fmpttd ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) → ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) : ( 𝐹𝑛 ) ⟶ ( wUniCl ‘ 𝐴 ) )
90 89 frnd ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) → ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ⊆ ( wUniCl ‘ 𝐴 ) )
91 83 90 unssd ( ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ∧ 𝑢 ∈ ( 𝐹𝑛 ) ) → ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ⊆ ( wUniCl ‘ 𝐴 ) )
92 91 ralrimiva ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) → ∀ 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ⊆ ( wUniCl ‘ 𝐴 ) )
93 iunss ( 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ⊆ ( wUniCl ‘ 𝐴 ) ↔ ∀ 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ⊆ ( wUniCl ‘ 𝐴 ) )
94 92 93 sylibr ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) → 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ⊆ ( wUniCl ‘ 𝐴 ) )
95 80 94 unssd ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) → ( ( ( 𝐹𝑛 ) ∪ ( 𝐹𝑛 ) ) ∪ 𝑢 ∈ ( 𝐹𝑛 ) ( { 𝒫 𝑢 , 𝑢 } ∪ ran ( 𝑣 ∈ ( 𝐹𝑛 ) ↦ { 𝑢 , 𝑣 } ) ) ) ⊆ ( wUniCl ‘ 𝐴 ) )
96 72 95 eqsstrd ( ( ( 𝐴𝑉𝑛 ∈ ω ) ∧ ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) → ( 𝐹 ‘ suc 𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) )
97 96 ex ( ( 𝐴𝑉𝑛 ∈ ω ) → ( ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) → ( 𝐹 ‘ suc 𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) )
98 97 expcom ( 𝑛 ∈ ω → ( 𝐴𝑉 → ( ( 𝐹𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) → ( 𝐹 ‘ suc 𝑛 ) ⊆ ( wUniCl ‘ 𝐴 ) ) ) )
99 13 15 17 32 98 finds2 ( 𝑚 ∈ ω → ( 𝐴𝑉 → ( 𝐹𝑚 ) ⊆ ( wUniCl ‘ 𝐴 ) ) )
100 99 com12 ( 𝐴𝑉 → ( 𝑚 ∈ ω → ( 𝐹𝑚 ) ⊆ ( wUniCl ‘ 𝐴 ) ) )
101 100 ralrimiv ( 𝐴𝑉 → ∀ 𝑚 ∈ ω ( 𝐹𝑚 ) ⊆ ( wUniCl ‘ 𝐴 ) )
102 iunss ( 𝑚 ∈ ω ( 𝐹𝑚 ) ⊆ ( wUniCl ‘ 𝐴 ) ↔ ∀ 𝑚 ∈ ω ( 𝐹𝑚 ) ⊆ ( wUniCl ‘ 𝐴 ) )
103 101 102 sylibr ( 𝐴𝑉 𝑚 ∈ ω ( 𝐹𝑚 ) ⊆ ( wUniCl ‘ 𝐴 ) )
104 11 103 eqsstrid ( 𝐴𝑉𝑈 ⊆ ( wUniCl ‘ 𝐴 ) )
105 5 104 eqssd ( 𝐴𝑉 → ( wUniCl ‘ 𝐴 ) = 𝑈 )