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Theorem wunex2 10763
Description: Construct a weak universe from a given set. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wunex2.f 𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)
wunex2.u 𝑈 = ran 𝐹
Assertion
Ref Expression
wunex2 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝑈(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem wunex2
Dummy variables 𝑢 𝑎 𝑣 𝑤 𝑏 𝑚 𝑛 𝑖 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wunex2.u . . . . . . . 8 𝑈 = ran 𝐹
21eleq2i 2817 . . . . . . 7 (𝑎𝑈𝑎 ran 𝐹)
3 frfnom 8456 . . . . . . . . 9 (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) Fn ω
4 wunex2.f . . . . . . . . . 10 𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)
54fneq1i 6652 . . . . . . . . 9 (𝐹 Fn ω ↔ (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) Fn ω)
63, 5mpbir 230 . . . . . . . 8 𝐹 Fn ω
7 fnunirn 7264 . . . . . . . 8 (𝐹 Fn ω → (𝑎 ran 𝐹 ↔ ∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚)))
86, 7ax-mp 5 . . . . . . 7 (𝑎 ran 𝐹 ↔ ∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚))
92, 8bitri 274 . . . . . 6 (𝑎𝑈 ↔ ∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚))
10 elssuni 4941 . . . . . . . . . . 11 (𝑎 ∈ (𝐹𝑚) → 𝑎 (𝐹𝑚))
1110ad2antll 727 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 (𝐹𝑚))
12 ssun2 4171 . . . . . . . . . . 11 (𝐹𝑚) ⊆ ((𝐹𝑚) ∪ (𝐹𝑚))
13 ssun1 4170 . . . . . . . . . . 11 ((𝐹𝑚) ∪ (𝐹𝑚)) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
1412, 13sstri 3986 . . . . . . . . . 10 (𝐹𝑚) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
1511, 14sstrdi 3989 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
16 simprl 769 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑚 ∈ ω)
17 fvex 6909 . . . . . . . . . . . 12 (𝐹𝑚) ∈ V
1817uniex 7747 . . . . . . . . . . . 12 (𝐹𝑚) ∈ V
1917, 18unex 7749 . . . . . . . . . . 11 ((𝐹𝑚) ∪ (𝐹𝑚)) ∈ V
20 prex 5434 . . . . . . . . . . . . 13 {𝒫 𝑢, 𝑢} ∈ V
2117mptex 7235 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) ∈ V
2221rnex 7918 . . . . . . . . . . . . 13 ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) ∈ V
2320, 22unex 7749 . . . . . . . . . . . 12 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ∈ V
2417, 23iunex 7973 . . . . . . . . . . 11 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ∈ V
2519, 24unex 7749 . . . . . . . . . 10 (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))) ∈ V
26 id 22 . . . . . . . . . . . . 13 (𝑤 = 𝑧𝑤 = 𝑧)
27 unieq 4920 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑤 = 𝑧)
2826, 27uneq12d 4161 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑤 𝑤) = (𝑧 𝑧))
29 pweq 4618 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
30 unieq 4920 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 𝑢 = 𝑥)
3129, 30preq12d 4747 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → {𝒫 𝑢, 𝑢} = {𝒫 𝑥, 𝑥})
32 preq2 4740 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑦 → {𝑢, 𝑣} = {𝑢, 𝑦})
3332cbvmptv 5262 . . . . . . . . . . . . . . . . 17 (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑦𝑤 ↦ {𝑢, 𝑦})
34 preq1 4739 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑥 → {𝑢, 𝑦} = {𝑥, 𝑦})
3534mpteq2dv 5251 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → (𝑦𝑤 ↦ {𝑢, 𝑦}) = (𝑦𝑤 ↦ {𝑥, 𝑦}))
3633, 35eqtrid 2777 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑦𝑤 ↦ {𝑥, 𝑦}))
3736rneqd 5940 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑦𝑤 ↦ {𝑥, 𝑦}))
3831, 37uneq12d 4161 . . . . . . . . . . . . . 14 (𝑢 = 𝑥 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦})))
3938cbviunv 5044 . . . . . . . . . . . . 13 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦}))
40 mpteq1 5242 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → (𝑦𝑤 ↦ {𝑥, 𝑦}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
4140rneqd 5940 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → ran (𝑦𝑤 ↦ {𝑥, 𝑦}) = ran (𝑦𝑧 ↦ {𝑥, 𝑦}))
4241uneq2d 4160 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 → ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
4326, 42iuneq12d 5025 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
4439, 43eqtrid 2777 . . . . . . . . . . . 12 (𝑤 = 𝑧 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
4528, 44uneq12d 4161 . . . . . . . . . . 11 (𝑤 = 𝑧 → ((𝑤 𝑤) ∪ 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣}))) = ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦}))))
46 id 22 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑚) → 𝑤 = (𝐹𝑚))
47 unieq 4920 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑚) → 𝑤 = (𝐹𝑚))
4846, 47uneq12d 4161 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑚) → (𝑤 𝑤) = ((𝐹𝑚) ∪ (𝐹𝑚)))
49 mpteq1 5242 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝑚) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))
5049rneqd 5940 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑚) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))
5150uneq2d 4160 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑚) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
5246, 51iuneq12d 5025 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑚) → 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
5348, 52uneq12d 4161 . . . . . . . . . . 11 (𝑤 = (𝐹𝑚) → ((𝑤 𝑤) ∪ 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣}))) = (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
544, 45, 53frsucmpt2 8461 . . . . . . . . . 10 ((𝑚 ∈ ω ∧ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))) ∈ V) → (𝐹‘suc 𝑚) = (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
5516, 25, 54sylancl 584 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (𝐹‘suc 𝑚) = (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
5615, 55sseqtrrd 4018 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 ⊆ (𝐹‘suc 𝑚))
57 fvssunirn 6929 . . . . . . . . 9 (𝐹‘suc 𝑚) ⊆ ran 𝐹
5857, 1sseqtrri 4014 . . . . . . . 8 (𝐹‘suc 𝑚) ⊆ 𝑈
5956, 58sstrdi 3989 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎𝑈)
6059rexlimdvaa 3145 . . . . . 6 (𝐴𝑉 → (∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚) → 𝑎𝑈))
619, 60biimtrid 241 . . . . 5 (𝐴𝑉 → (𝑎𝑈𝑎𝑈))
6261ralrimiv 3134 . . . 4 (𝐴𝑉 → ∀𝑎𝑈 𝑎𝑈)
63 dftr3 5272 . . . 4 (Tr 𝑈 ↔ ∀𝑎𝑈 𝑎𝑈)
6462, 63sylibr 233 . . 3 (𝐴𝑉 → Tr 𝑈)
65 1on 8499 . . . . . . . 8 1o ∈ On
66 unexg 7752 . . . . . . . 8 ((𝐴𝑉 ∧ 1o ∈ On) → (𝐴 ∪ 1o) ∈ V)
6765, 66mpan2 689 . . . . . . 7 (𝐴𝑉 → (𝐴 ∪ 1o) ∈ V)
684fveq1i 6897 . . . . . . . 8 (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅)
69 fr0g 8457 . . . . . . . 8 ((𝐴 ∪ 1o) ∈ V → ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅) = (𝐴 ∪ 1o))
7068, 69eqtrid 2777 . . . . . . 7 ((𝐴 ∪ 1o) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1o))
7167, 70syl 17 . . . . . 6 (𝐴𝑉 → (𝐹‘∅) = (𝐴 ∪ 1o))
72 fvssunirn 6929 . . . . . . 7 (𝐹‘∅) ⊆ ran 𝐹
7372, 1sseqtrri 4014 . . . . . 6 (𝐹‘∅) ⊆ 𝑈
7471, 73eqsstrrdi 4032 . . . . 5 (𝐴𝑉 → (𝐴 ∪ 1o) ⊆ 𝑈)
7574unssbd 4186 . . . 4 (𝐴𝑉 → 1o𝑈)
76 1n0 8509 . . . 4 1o ≠ ∅
77 ssn0 4402 . . . 4 ((1o𝑈 ∧ 1o ≠ ∅) → 𝑈 ≠ ∅)
7875, 76, 77sylancl 584 . . 3 (𝐴𝑉𝑈 ≠ ∅)
79 pweq 4618 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → 𝒫 𝑢 = 𝒫 𝑎)
80 unieq 4920 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 𝑢 = 𝑎)
8179, 80preq12d 4747 . . . . . . . . . . . . . 14 (𝑢 = 𝑎 → {𝒫 𝑢, 𝑢} = {𝒫 𝑎, 𝑎})
82 preq1 4739 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑎 → {𝑢, 𝑣} = {𝑎, 𝑣})
8382mpteq2dv 5251 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣}))
8483rneqd 5940 . . . . . . . . . . . . . 14 (𝑢 = 𝑎 → ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣}))
8581, 84uneq12d 4161 . . . . . . . . . . . . 13 (𝑢 = 𝑎 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) = ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣})))
8685ssiun2s 5052 . . . . . . . . . . . 12 (𝑎 ∈ (𝐹𝑚) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣})) ⊆ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
8786ad2antll 727 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣})) ⊆ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
88 ssun2 4171 . . . . . . . . . . . . 13 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
8988, 55sseqtrrid 4030 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ⊆ (𝐹‘suc 𝑚))
9089, 58sstrdi 3989 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ⊆ 𝑈)
9187, 90sstrd 3987 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣})) ⊆ 𝑈)
9291unssad 4185 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → {𝒫 𝑎, 𝑎} ⊆ 𝑈)
93 vpwex 5377 . . . . . . . . . 10 𝒫 𝑎 ∈ V
94 vuniex 7745 . . . . . . . . . 10 𝑎 ∈ V
9593, 94prss 4825 . . . . . . . . 9 ((𝒫 𝑎𝑈 𝑎𝑈) ↔ {𝒫 𝑎, 𝑎} ⊆ 𝑈)
9692, 95sylibr 233 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (𝒫 𝑎𝑈 𝑎𝑈))
9796simprd 494 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎𝑈)
9896simpld 493 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝒫 𝑎𝑈)
991eleq2i 2817 . . . . . . . . . 10 (𝑏𝑈𝑏 ran 𝐹)
100 fnunirn 7264 . . . . . . . . . . 11 (𝐹 Fn ω → (𝑏 ran 𝐹 ↔ ∃𝑛 ∈ ω 𝑏 ∈ (𝐹𝑛)))
1016, 100ax-mp 5 . . . . . . . . . 10 (𝑏 ran 𝐹 ↔ ∃𝑛 ∈ ω 𝑏 ∈ (𝐹𝑛))
10299, 101bitri 274 . . . . . . . . 9 (𝑏𝑈 ↔ ∃𝑛 ∈ ω 𝑏 ∈ (𝐹𝑛))
103 ordom 7881 . . . . . . . . . . . . . . . . 17 Ord ω
104 simplrl 775 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑚 ∈ ω)
105 simprl 769 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑛 ∈ ω)
106 ordunel 7831 . . . . . . . . . . . . . . . . 17 ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
107103, 104, 105, 106mp3an2i 1462 . . . . . . . . . . . . . . . 16 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → (𝑚𝑛) ∈ ω)
108 ssun1 4170 . . . . . . . . . . . . . . . . 17 𝑚 ⊆ (𝑚𝑛)
109 fveq2 6896 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
110109sseq2d 4009 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹𝑚)))
111 fveq2 6896 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑖 → (𝐹𝑘) = (𝐹𝑖))
112111sseq2d 4009 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑖 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹𝑖)))
113 fveq2 6896 . . . . . . . . . . . . . . . . . . 19 (𝑘 = suc 𝑖 → (𝐹𝑘) = (𝐹‘suc 𝑖))
114113sseq2d 4009 . . . . . . . . . . . . . . . . . 18 (𝑘 = suc 𝑖 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹‘suc 𝑖)))
115 fveq2 6896 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑚𝑛) → (𝐹𝑘) = (𝐹‘(𝑚𝑛)))
116115sseq2d 4009 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑚𝑛) → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛))))
117 ssidd 4000 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ω → (𝐹𝑚) ⊆ (𝐹𝑚))
118 fveq2 6896 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑖 → (𝐹𝑚) = (𝐹𝑖))
119 suceq 6437 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑖 → suc 𝑚 = suc 𝑖)
120119fveq2d 6900 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑖 → (𝐹‘suc 𝑚) = (𝐹‘suc 𝑖))
121118, 120sseq12d 4010 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑖 → ((𝐹𝑚) ⊆ (𝐹‘suc 𝑚) ↔ (𝐹𝑖) ⊆ (𝐹‘suc 𝑖)))
122 ssun1 4170 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹𝑚) ⊆ ((𝐹𝑚) ∪ (𝐹𝑚))
123122, 13sstri 3986 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹𝑚) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
12425, 54mpan2 689 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ω → (𝐹‘suc 𝑚) = (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
125123, 124sseqtrrid 4030 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ω → (𝐹𝑚) ⊆ (𝐹‘suc 𝑚))
126121, 125vtoclga 3556 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ω → (𝐹𝑖) ⊆ (𝐹‘suc 𝑖))
127126ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑖) ⊆ (𝐹‘suc 𝑖))
128 sstr2 3983 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑚) ⊆ (𝐹𝑖) → ((𝐹𝑖) ⊆ (𝐹‘suc 𝑖) → (𝐹𝑚) ⊆ (𝐹‘suc 𝑖)))
129127, 128syl5com 31 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → ((𝐹𝑚) ⊆ (𝐹𝑖) → (𝐹𝑚) ⊆ (𝐹‘suc 𝑖)))
130110, 112, 114, 116, 117, 129findsg 7905 . . . . . . . . . . . . . . . . 17 ((((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚 ⊆ (𝑚𝑛)) → (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛)))
131108, 130mpan2 689 . . . . . . . . . . . . . . . 16 (((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω) → (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛)))
132107, 104, 131syl2anc 582 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛)))
133 simplrr 776 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑎 ∈ (𝐹𝑚))
134132, 133sseldd 3977 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑎 ∈ (𝐹‘(𝑚𝑛)))
13582mpteq2dv 5251 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑎 → (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
136135rneqd 5940 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑎 → ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
13781, 136uneq12d 4161 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) = ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})))
138137ssiun2s 5052 . . . . . . . . . . . . . 14 (𝑎 ∈ (𝐹‘(𝑚𝑛)) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})) ⊆ 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})))
139134, 138syl 17 . . . . . . . . . . . . 13 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})) ⊆ 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})))
140 ssun2 4171 . . . . . . . . . . . . . . 15 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ⊆ (((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))) ∪ 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})))
141 fvex 6909 . . . . . . . . . . . . . . . . . 18 (𝐹‘(𝑚𝑛)) ∈ V
142141uniex 7747 . . . . . . . . . . . . . . . . . 18 (𝐹‘(𝑚𝑛)) ∈ V
143141, 142unex 7749 . . . . . . . . . . . . . . . . 17 ((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))) ∈ V
144141mptex 7235 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) ∈ V
145144rnex 7918 . . . . . . . . . . . . . . . . . . 19 ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) ∈ V
14620, 145unex 7749 . . . . . . . . . . . . . . . . . 18 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ∈ V
147141, 146iunex 7973 . . . . . . . . . . . . . . . . 17 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ∈ V
148143, 147unex 7749 . . . . . . . . . . . . . . . 16 (((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))) ∪ 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))) ∈ V
149 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹‘(𝑚𝑛)) → 𝑤 = (𝐹‘(𝑚𝑛)))
150 unieq 4920 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹‘(𝑚𝑛)) → 𝑤 = (𝐹‘(𝑚𝑛)))
151149, 150uneq12d 4161 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝐹‘(𝑚𝑛)) → (𝑤 𝑤) = ((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))))
152 mpteq1 5242 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐹‘(𝑚𝑛)) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))
153152rneqd 5940 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐹‘(𝑚𝑛)) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))
154153uneq2d 4160 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹‘(𝑚𝑛)) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})))
155149, 154iuneq12d 5025 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝐹‘(𝑚𝑛)) → 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})))
156151, 155uneq12d 4161 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝐹‘(𝑚𝑛)) → ((𝑤 𝑤) ∪ 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣}))) = (((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))) ∪ 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))))
1574, 45, 156frsucmpt2 8461 . . . . . . . . . . . . . . . 16 (((𝑚𝑛) ∈ ω ∧ (((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))) ∪ 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))) ∈ V) → (𝐹‘suc (𝑚𝑛)) = (((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))) ∪ 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))))
158107, 148, 157sylancl 584 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → (𝐹‘suc (𝑚𝑛)) = (((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))) ∪ 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))))
159140, 158sseqtrrid 4030 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ⊆ (𝐹‘suc (𝑚𝑛)))
160 fvssunirn 6929 . . . . . . . . . . . . . . 15 (𝐹‘suc (𝑚𝑛)) ⊆ ran 𝐹
161160, 1sseqtrri 4014 . . . . . . . . . . . . . 14 (𝐹‘suc (𝑚𝑛)) ⊆ 𝑈
162159, 161sstrdi 3989 . . . . . . . . . . . . 13 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ⊆ 𝑈)
163139, 162sstrd 3987 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})) ⊆ 𝑈)
164163unssbd 4186 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}) ⊆ 𝑈)
165 ssun2 4171 . . . . . . . . . . . . . . . . . . 19 𝑛 ⊆ (𝑚𝑛)
166 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑖 = (𝑚𝑛) → 𝑖 = (𝑚𝑛))
167165, 166sseqtrrid 4030 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑚𝑛) → 𝑛𝑖)
168167biantrud 530 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚𝑛) → (𝑛 ∈ ω ↔ (𝑛 ∈ ω ∧ 𝑛𝑖)))
169168bicomd 222 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚𝑛) → ((𝑛 ∈ ω ∧ 𝑛𝑖) ↔ 𝑛 ∈ ω))
170 fveq2 6896 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚𝑛) → (𝐹𝑖) = (𝐹‘(𝑚𝑛)))
171170sseq2d 4009 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚𝑛) → ((𝐹𝑛) ⊆ (𝐹𝑖) ↔ (𝐹𝑛) ⊆ (𝐹‘(𝑚𝑛))))
172169, 171imbi12d 343 . . . . . . . . . . . . . . 15 (𝑖 = (𝑚𝑛) → (((𝑛 ∈ ω ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖)) ↔ (𝑛 ∈ ω → (𝐹𝑛) ⊆ (𝐹‘(𝑚𝑛)))))
173 eleq1w 2808 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑚 ∈ ω ↔ 𝑛 ∈ ω))
174173anbi2d 628 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → ((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ↔ (𝑖 ∈ ω ∧ 𝑛 ∈ ω)))
175 sseq1 4002 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → (𝑚𝑖𝑛𝑖))
176174, 175anbi12d 630 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) ↔ ((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖)))
177 fveq2 6896 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
178177sseq1d 4008 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → ((𝐹𝑚) ⊆ (𝐹𝑖) ↔ (𝐹𝑛) ⊆ (𝐹𝑖)))
179176, 178imbi12d 343 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → ((((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑚) ⊆ (𝐹𝑖)) ↔ (((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖))))
180110, 112, 114, 112, 117, 129findsg 7905 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑚) ⊆ (𝐹𝑖))
181179, 180chvarvv 1994 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖))
182181expl 456 . . . . . . . . . . . . . . 15 (𝑖 ∈ ω → ((𝑛 ∈ ω ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖)))
183172, 182vtoclga 3556 . . . . . . . . . . . . . 14 ((𝑚𝑛) ∈ ω → (𝑛 ∈ ω → (𝐹𝑛) ⊆ (𝐹‘(𝑚𝑛))))
184107, 105, 183sylc 65 . . . . . . . . . . . . 13 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → (𝐹𝑛) ⊆ (𝐹‘(𝑚𝑛)))
185 simprr 771 . . . . . . . . . . . . 13 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑏 ∈ (𝐹𝑛))
186184, 185sseldd 3977 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑏 ∈ (𝐹‘(𝑚𝑛)))
187 prex 5434 . . . . . . . . . . . 12 {𝑎, 𝑏} ∈ V
188 eqid 2725 . . . . . . . . . . . . 13 (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})
189 preq2 4740 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → {𝑎, 𝑣} = {𝑎, 𝑏})
190188, 189elrnmpt1s 5959 . . . . . . . . . . . 12 ((𝑏 ∈ (𝐹‘(𝑚𝑛)) ∧ {𝑎, 𝑏} ∈ V) → {𝑎, 𝑏} ∈ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
191186, 187, 190sylancl 584 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → {𝑎, 𝑏} ∈ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
192164, 191sseldd 3977 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → {𝑎, 𝑏} ∈ 𝑈)
193192rexlimdvaa 3145 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (∃𝑛 ∈ ω 𝑏 ∈ (𝐹𝑛) → {𝑎, 𝑏} ∈ 𝑈))
194102, 193biimtrid 241 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (𝑏𝑈 → {𝑎, 𝑏} ∈ 𝑈))
195194ralrimiv 3134 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)
19697, 98, 1953jca 1125 . . . . . 6 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈))
197196rexlimdvaa 3145 . . . . 5 (𝐴𝑉 → (∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚) → ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)))
1989, 197biimtrid 241 . . . 4 (𝐴𝑉 → (𝑎𝑈 → ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)))
199198ralrimiv 3134 . . 3 (𝐴𝑉 → ∀𝑎𝑈 ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈))
200 rdgfun 8437 . . . . . . . . 9 Fun rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o))
201 omex 9668 . . . . . . . . 9 ω ∈ V
202 resfunexg 7227 . . . . . . . . 9 ((Fun rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ∧ ω ∈ V) → (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) ∈ V)
203200, 201, 202mp2an 690 . . . . . . . 8 (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) ∈ V
2044, 203eqeltri 2821 . . . . . . 7 𝐹 ∈ V
205204rnex 7918 . . . . . 6 ran 𝐹 ∈ V
206205uniex 7747 . . . . 5 ran 𝐹 ∈ V
2071, 206eqeltri 2821 . . . 4 𝑈 ∈ V
208 iswun 10729 . . . 4 (𝑈 ∈ V → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑎𝑈 ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈))))
209207, 208ax-mp 5 . . 3 (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑎𝑈 ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)))
21064, 78, 199, 209syl3anbrc 1340 . 2 (𝐴𝑉𝑈 ∈ WUni)
21174unssad 4185 . 2 (𝐴𝑉𝐴𝑈)
212210, 211jca 510 1 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  Vcvv 3461  cun 3942  wss 3944  c0 4322  𝒫 cpw 4604  {cpr 4632   cuni 4909   ciun 4997  cmpt 5232  Tr wtr 5266  ran crn 5679  cres 5680  Ord word 6370  Oncon0 6371  suc csuc 6373  Fun wfun 6543   Fn wfn 6544  cfv 6549  ωcom 7871  reccrdg 8430  1oc1o 8480  WUnicwun 10725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-wun 10727
This theorem is referenced by:  wunex  10764  wuncval2  10772
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