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Theorem wunex2 10775
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 2830 . . . . . . 7 (𝑎𝑈𝑎 ran 𝐹)
3 frfnom 8473 . . . . . . . . 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 6665 . . . . . . . . 9 (𝐹 Fn ω ↔ (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) Fn ω)
63, 5mpbir 231 . . . . . . . 8 𝐹 Fn ω
7 fnunirn 7273 . . . . . . . 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 275 . . . . . 6 (𝑎𝑈 ↔ ∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚))
10 elssuni 4941 . . . . . . . . . . 11 (𝑎 ∈ (𝐹𝑚) → 𝑎 (𝐹𝑚))
1110ad2antll 729 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 (𝐹𝑚))
12 ssun2 4188 . . . . . . . . . . 11 (𝐹𝑚) ⊆ ((𝐹𝑚) ∪ (𝐹𝑚))
13 ssun1 4187 . . . . . . . . . . 11 ((𝐹𝑚) ∪ (𝐹𝑚)) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
1412, 13sstri 4004 . . . . . . . . . 10 (𝐹𝑚) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
1511, 14sstrdi 4007 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
16 simprl 771 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑚 ∈ ω)
17 fvex 6919 . . . . . . . . . . . 12 (𝐹𝑚) ∈ V
1817uniex 7759 . . . . . . . . . . . 12 (𝐹𝑚) ∈ V
1917, 18unex 7762 . . . . . . . . . . 11 ((𝐹𝑚) ∪ (𝐹𝑚)) ∈ V
20 prex 5442 . . . . . . . . . . . . 13 {𝒫 𝑢, 𝑢} ∈ V
2117mptex 7242 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) ∈ V
2221rnex 7932 . . . . . . . . . . . . 13 ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) ∈ V
2320, 22unex 7762 . . . . . . . . . . . 12 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ∈ V
2417, 23iunex 7991 . . . . . . . . . . 11 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ∈ V
2519, 24unex 7762 . . . . . . . . . 10 (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))) ∈ V
26 id 22 . . . . . . . . . . . . 13 (𝑤 = 𝑧𝑤 = 𝑧)
27 unieq 4922 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑤 = 𝑧)
2826, 27uneq12d 4178 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑤 𝑤) = (𝑧 𝑧))
29 pweq 4618 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
30 unieq 4922 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 𝑢 = 𝑥)
3129, 30preq12d 4745 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → {𝒫 𝑢, 𝑢} = {𝒫 𝑥, 𝑥})
32 preq2 4738 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑦 → {𝑢, 𝑣} = {𝑢, 𝑦})
3332cbvmptv 5260 . . . . . . . . . . . . . . . . 17 (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑦𝑤 ↦ {𝑢, 𝑦})
34 preq1 4737 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑥 → {𝑢, 𝑦} = {𝑥, 𝑦})
3534mpteq2dv 5249 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → (𝑦𝑤 ↦ {𝑢, 𝑦}) = (𝑦𝑤 ↦ {𝑥, 𝑦}))
3633, 35eqtrid 2786 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑦𝑤 ↦ {𝑥, 𝑦}))
3736rneqd 5951 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑦𝑤 ↦ {𝑥, 𝑦}))
3831, 37uneq12d 4178 . . . . . . . . . . . . . 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 5240 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → (𝑦𝑤 ↦ {𝑥, 𝑦}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
4140rneqd 5951 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → ran (𝑦𝑤 ↦ {𝑥, 𝑦}) = ran (𝑦𝑧 ↦ {𝑥, 𝑦}))
4241uneq2d 4177 . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . 12 (𝑤 = 𝑧 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
4528, 44uneq12d 4178 . . . . . . . . . . 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 4922 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑚) → 𝑤 = (𝐹𝑚))
4846, 47uneq12d 4178 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑚) → (𝑤 𝑤) = ((𝐹𝑚) ∪ (𝐹𝑚)))
49 mpteq1 5240 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝑚) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))
5049rneqd 5951 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑚) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))
5150uneq2d 4177 . . . . . . . . . . . . 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 4178 . . . . . . . . . . 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 8478 . . . . . . . . . 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 586 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (𝐹‘suc 𝑚) = (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
5615, 55sseqtrrd 4036 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 ⊆ (𝐹‘suc 𝑚))
57 fvssunirn 6939 . . . . . . . . 9 (𝐹‘suc 𝑚) ⊆ ran 𝐹
5857, 1sseqtrri 4032 . . . . . . . 8 (𝐹‘suc 𝑚) ⊆ 𝑈
5956, 58sstrdi 4007 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎𝑈)
6059rexlimdvaa 3153 . . . . . 6 (𝐴𝑉 → (∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚) → 𝑎𝑈))
619, 60biimtrid 242 . . . . 5 (𝐴𝑉 → (𝑎𝑈𝑎𝑈))
6261ralrimiv 3142 . . . 4 (𝐴𝑉 → ∀𝑎𝑈 𝑎𝑈)
63 dftr3 5270 . . . 4 (Tr 𝑈 ↔ ∀𝑎𝑈 𝑎𝑈)
6462, 63sylibr 234 . . 3 (𝐴𝑉 → Tr 𝑈)
65 1on 8516 . . . . . . . 8 1o ∈ On
66 unexg 7761 . . . . . . . 8 ((𝐴𝑉 ∧ 1o ∈ On) → (𝐴 ∪ 1o) ∈ V)
6765, 66mpan2 691 . . . . . . 7 (𝐴𝑉 → (𝐴 ∪ 1o) ∈ V)
684fveq1i 6907 . . . . . . . 8 (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅)
69 fr0g 8474 . . . . . . . 8 ((𝐴 ∪ 1o) ∈ V → ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅) = (𝐴 ∪ 1o))
7068, 69eqtrid 2786 . . . . . . 7 ((𝐴 ∪ 1o) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1o))
7167, 70syl 17 . . . . . 6 (𝐴𝑉 → (𝐹‘∅) = (𝐴 ∪ 1o))
72 fvssunirn 6939 . . . . . . 7 (𝐹‘∅) ⊆ ran 𝐹
7372, 1sseqtrri 4032 . . . . . 6 (𝐹‘∅) ⊆ 𝑈
7471, 73eqsstrrdi 4050 . . . . 5 (𝐴𝑉 → (𝐴 ∪ 1o) ⊆ 𝑈)
7574unssbd 4203 . . . 4 (𝐴𝑉 → 1o𝑈)
76 1n0 8524 . . . 4 1o ≠ ∅
77 ssn0 4409 . . . 4 ((1o𝑈 ∧ 1o ≠ ∅) → 𝑈 ≠ ∅)
7875, 76, 77sylancl 586 . . 3 (𝐴𝑉𝑈 ≠ ∅)
79 pweq 4618 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → 𝒫 𝑢 = 𝒫 𝑎)
80 unieq 4922 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 𝑢 = 𝑎)
8179, 80preq12d 4745 . . . . . . . . . . . . . 14 (𝑢 = 𝑎 → {𝒫 𝑢, 𝑢} = {𝒫 𝑎, 𝑎})
82 preq1 4737 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑎 → {𝑢, 𝑣} = {𝑎, 𝑣})
8382mpteq2dv 5249 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣}))
8483rneqd 5951 . . . . . . . . . . . . . 14 (𝑢 = 𝑎 → ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣}))
8581, 84uneq12d 4178 . . . . . . . . . . . . 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 729 . . . . . . . . . . 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 4188 . . . . . . . . . . . . 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 4048 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ⊆ (𝐹‘suc 𝑚))
9089, 58sstrdi 4007 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ⊆ 𝑈)
9187, 90sstrd 4005 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣})) ⊆ 𝑈)
9291unssad 4202 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → {𝒫 𝑎, 𝑎} ⊆ 𝑈)
93 vpwex 5382 . . . . . . . . . 10 𝒫 𝑎 ∈ V
94 vuniex 7757 . . . . . . . . . 10 𝑎 ∈ V
9593, 94prss 4824 . . . . . . . . 9 ((𝒫 𝑎𝑈 𝑎𝑈) ↔ {𝒫 𝑎, 𝑎} ⊆ 𝑈)
9692, 95sylibr 234 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (𝒫 𝑎𝑈 𝑎𝑈))
9796simprd 495 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎𝑈)
9896simpld 494 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝒫 𝑎𝑈)
991eleq2i 2830 . . . . . . . . . 10 (𝑏𝑈𝑏 ran 𝐹)
100 fnunirn 7273 . . . . . . . . . . 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 275 . . . . . . . . 9 (𝑏𝑈 ↔ ∃𝑛 ∈ ω 𝑏 ∈ (𝐹𝑛))
103 ordom 7896 . . . . . . . . . . . . . . . . 17 Ord ω
104 simplrl 777 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑚 ∈ ω)
105 simprl 771 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑛 ∈ ω)
106 ordunel 7846 . . . . . . . . . . . . . . . . 17 ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
107103, 104, 105, 106mp3an2i 1465 . . . . . . . . . . . . . . . 16 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → (𝑚𝑛) ∈ ω)
108 ssun1 4187 . . . . . . . . . . . . . . . . 17 𝑚 ⊆ (𝑚𝑛)
109 fveq2 6906 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
110109sseq2d 4027 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹𝑚)))
111 fveq2 6906 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑖 → (𝐹𝑘) = (𝐹𝑖))
112111sseq2d 4027 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑖 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹𝑖)))
113 fveq2 6906 . . . . . . . . . . . . . . . . . . 19 (𝑘 = suc 𝑖 → (𝐹𝑘) = (𝐹‘suc 𝑖))
114113sseq2d 4027 . . . . . . . . . . . . . . . . . 18 (𝑘 = suc 𝑖 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹‘suc 𝑖)))
115 fveq2 6906 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑚𝑛) → (𝐹𝑘) = (𝐹‘(𝑚𝑛)))
116115sseq2d 4027 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑚𝑛) → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛))))
117 ssidd 4018 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ω → (𝐹𝑚) ⊆ (𝐹𝑚))
118 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑖 → (𝐹𝑚) = (𝐹𝑖))
119 suceq 6451 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑖 → suc 𝑚 = suc 𝑖)
120119fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑖 → (𝐹‘suc 𝑚) = (𝐹‘suc 𝑖))
121118, 120sseq12d 4028 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑖 → ((𝐹𝑚) ⊆ (𝐹‘suc 𝑚) ↔ (𝐹𝑖) ⊆ (𝐹‘suc 𝑖)))
122 ssun1 4187 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹𝑚) ⊆ ((𝐹𝑚) ∪ (𝐹𝑚))
123122, 13sstri 4004 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹𝑚) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
12425, 54mpan2 691 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ω → (𝐹‘suc 𝑚) = (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
125123, 124sseqtrrid 4048 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ω → (𝐹𝑚) ⊆ (𝐹‘suc 𝑚))
126121, 125vtoclga 3576 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ω → (𝐹𝑖) ⊆ (𝐹‘suc 𝑖))
127126ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑖) ⊆ (𝐹‘suc 𝑖))
128 sstr2 4001 . . . . . . . . . . . . . . . . . . 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 7919 . . . . . . . . . . . . . . . . 17 ((((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚 ⊆ (𝑚𝑛)) → (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛)))
131108, 130mpan2 691 . . . . . . . . . . . . . . . 16 (((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω) → (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛)))
132107, 104, 131syl2anc 584 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛)))
133 simplrr 778 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑎 ∈ (𝐹𝑚))
134132, 133sseldd 3995 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑎 ∈ (𝐹‘(𝑚𝑛)))
13582mpteq2dv 5249 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑎 → (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
136135rneqd 5951 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑎 → ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
13781, 136uneq12d 4178 . . . . . . . . . . . . . . 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 4188 . . . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . . . . . 18 (𝐹‘(𝑚𝑛)) ∈ V
142141uniex 7759 . . . . . . . . . . . . . . . . . 18 (𝐹‘(𝑚𝑛)) ∈ V
143141, 142unex 7762 . . . . . . . . . . . . . . . . 17 ((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))) ∈ V
144141mptex 7242 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) ∈ V
145144rnex 7932 . . . . . . . . . . . . . . . . . . 19 ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) ∈ V
14620, 145unex 7762 . . . . . . . . . . . . . . . . . 18 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ∈ V
147141, 146iunex 7991 . . . . . . . . . . . . . . . . 17 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ∈ V
148143, 147unex 7762 . . . . . . . . . . . . . . . 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 4922 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹‘(𝑚𝑛)) → 𝑤 = (𝐹‘(𝑚𝑛)))
151149, 150uneq12d 4178 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝐹‘(𝑚𝑛)) → (𝑤 𝑤) = ((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))))
152 mpteq1 5240 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐹‘(𝑚𝑛)) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))
153152rneqd 5951 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐹‘(𝑚𝑛)) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))
154153uneq2d 4177 . . . . . . . . . . . . . . . . . . 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 4178 . . . . . . . . . . . . . . . . 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 8478 . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . 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 4048 . . . . . . . . . . . . . 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 6939 . . . . . . . . . . . . . . 15 (𝐹‘suc (𝑚𝑛)) ⊆ ran 𝐹
161160, 1sseqtrri 4032 . . . . . . . . . . . . . 14 (𝐹‘suc (𝑚𝑛)) ⊆ 𝑈
162159, 161sstrdi 4007 . . . . . . . . . . . . 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 4005 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})) ⊆ 𝑈)
164163unssbd 4203 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}) ⊆ 𝑈)
165 ssun2 4188 . . . . . . . . . . . . . . . . . . 19 𝑛 ⊆ (𝑚𝑛)
166 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑖 = (𝑚𝑛) → 𝑖 = (𝑚𝑛))
167165, 166sseqtrrid 4048 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑚𝑛) → 𝑛𝑖)
168167biantrud 531 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚𝑛) → (𝑛 ∈ ω ↔ (𝑛 ∈ ω ∧ 𝑛𝑖)))
169168bicomd 223 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚𝑛) → ((𝑛 ∈ ω ∧ 𝑛𝑖) ↔ 𝑛 ∈ ω))
170 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚𝑛) → (𝐹𝑖) = (𝐹‘(𝑚𝑛)))
171170sseq2d 4027 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚𝑛) → ((𝐹𝑛) ⊆ (𝐹𝑖) ↔ (𝐹𝑛) ⊆ (𝐹‘(𝑚𝑛))))
172169, 171imbi12d 344 . . . . . . . . . . . . . . 15 (𝑖 = (𝑚𝑛) → (((𝑛 ∈ ω ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖)) ↔ (𝑛 ∈ ω → (𝐹𝑛) ⊆ (𝐹‘(𝑚𝑛)))))
173 eleq1w 2821 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑚 ∈ ω ↔ 𝑛 ∈ ω))
174173anbi2d 630 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → ((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ↔ (𝑖 ∈ ω ∧ 𝑛 ∈ ω)))
175 sseq1 4020 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → (𝑚𝑖𝑛𝑖))
176174, 175anbi12d 632 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) ↔ ((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖)))
177 fveq2 6906 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
178177sseq1d 4026 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → ((𝐹𝑚) ⊆ (𝐹𝑖) ↔ (𝐹𝑛) ⊆ (𝐹𝑖)))
179176, 178imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → ((((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑚) ⊆ (𝐹𝑖)) ↔ (((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖))))
180110, 112, 114, 112, 117, 129findsg 7919 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑚) ⊆ (𝐹𝑖))
181179, 180chvarvv 1995 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖))
182181expl 457 . . . . . . . . . . . . . . 15 (𝑖 ∈ ω → ((𝑛 ∈ ω ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖)))
183172, 182vtoclga 3576 . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . 13 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑏 ∈ (𝐹𝑛))
186184, 185sseldd 3995 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑏 ∈ (𝐹‘(𝑚𝑛)))
187 prex 5442 . . . . . . . . . . . 12 {𝑎, 𝑏} ∈ V
188 eqid 2734 . . . . . . . . . . . . 13 (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})
189 preq2 4738 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → {𝑎, 𝑣} = {𝑎, 𝑏})
190188, 189elrnmpt1s 5972 . . . . . . . . . . . 12 ((𝑏 ∈ (𝐹‘(𝑚𝑛)) ∧ {𝑎, 𝑏} ∈ V) → {𝑎, 𝑏} ∈ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
191186, 187, 190sylancl 586 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → {𝑎, 𝑏} ∈ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
192164, 191sseldd 3995 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → {𝑎, 𝑏} ∈ 𝑈)
193192rexlimdvaa 3153 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (∃𝑛 ∈ ω 𝑏 ∈ (𝐹𝑛) → {𝑎, 𝑏} ∈ 𝑈))
194102, 193biimtrid 242 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (𝑏𝑈 → {𝑎, 𝑏} ∈ 𝑈))
195194ralrimiv 3142 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)
19697, 98, 1953jca 1127 . . . . . 6 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈))
197196rexlimdvaa 3153 . . . . 5 (𝐴𝑉 → (∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚) → ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)))
1989, 197biimtrid 242 . . . 4 (𝐴𝑉 → (𝑎𝑈 → ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)))
199198ralrimiv 3142 . . 3 (𝐴𝑉 → ∀𝑎𝑈 ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈))
200 rdgfun 8454 . . . . . . . . 9 Fun rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o))
201 omex 9680 . . . . . . . . 9 ω ∈ V
202 resfunexg 7234 . . . . . . . . 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 692 . . . . . . . 8 (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) ∈ V
2044, 203eqeltri 2834 . . . . . . 7 𝐹 ∈ V
205204rnex 7932 . . . . . 6 ran 𝐹 ∈ V
206205uniex 7759 . . . . 5 ran 𝐹 ∈ V
2071, 206eqeltri 2834 . . . 4 𝑈 ∈ V
208 iswun 10741 . . . 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 1342 . 2 (𝐴𝑉𝑈 ∈ WUni)
21174unssad 4202 . 2 (𝐴𝑉𝐴𝑈)
212210, 211jca 511 1 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  Vcvv 3477  cun 3960  wss 3962  c0 4338  𝒫 cpw 4604  {cpr 4632   cuni 4911   ciun 4995  cmpt 5230  Tr wtr 5264  ran crn 5689  cres 5690  Ord word 6384  Oncon0 6385  suc csuc 6387  Fun wfun 6556   Fn wfn 6557  cfv 6562  ωcom 7886  reccrdg 8447  1oc1o 8497  WUnicwun 10737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-wun 10739
This theorem is referenced by:  wunex  10776  wuncval2  10784
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