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Theorem wunex2 10674
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 2829 . . . . . . 7 (𝑎𝑈𝑎 ran 𝐹)
3 frfnom 8381 . . . . . . . . 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 6599 . . . . . . . . 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 7201 . . . . . . . 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 4898 . . . . . . . . . . 11 (𝑎 ∈ (𝐹𝑚) → 𝑎 (𝐹𝑚))
1110ad2antll 727 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 (𝐹𝑚))
12 ssun2 4133 . . . . . . . . . . 11 (𝐹𝑚) ⊆ ((𝐹𝑚) ∪ (𝐹𝑚))
13 ssun1 4132 . . . . . . . . . . 11 ((𝐹𝑚) ∪ (𝐹𝑚)) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
1412, 13sstri 3953 . . . . . . . . . 10 (𝐹𝑚) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
1511, 14sstrdi 3956 . . . . . . . . 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 6855 . . . . . . . . . . . 12 (𝐹𝑚) ∈ V
1817uniex 7678 . . . . . . . . . . . 12 (𝐹𝑚) ∈ V
1917, 18unex 7680 . . . . . . . . . . 11 ((𝐹𝑚) ∪ (𝐹𝑚)) ∈ V
20 prex 5389 . . . . . . . . . . . . 13 {𝒫 𝑢, 𝑢} ∈ V
2117mptex 7173 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) ∈ V
2221rnex 7849 . . . . . . . . . . . . 13 ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) ∈ V
2320, 22unex 7680 . . . . . . . . . . . 12 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ∈ V
2417, 23iunex 7901 . . . . . . . . . . 11 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ∈ V
2519, 24unex 7680 . . . . . . . . . 10 (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))) ∈ V
26 id 22 . . . . . . . . . . . . 13 (𝑤 = 𝑧𝑤 = 𝑧)
27 unieq 4876 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑤 = 𝑧)
2826, 27uneq12d 4124 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑤 𝑤) = (𝑧 𝑧))
29 pweq 4574 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
30 unieq 4876 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 𝑢 = 𝑥)
3129, 30preq12d 4702 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → {𝒫 𝑢, 𝑢} = {𝒫 𝑥, 𝑥})
32 preq2 4695 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑦 → {𝑢, 𝑣} = {𝑢, 𝑦})
3332cbvmptv 5218 . . . . . . . . . . . . . . . . 17 (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑦𝑤 ↦ {𝑢, 𝑦})
34 preq1 4694 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑥 → {𝑢, 𝑦} = {𝑥, 𝑦})
3534mpteq2dv 5207 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → (𝑦𝑤 ↦ {𝑢, 𝑦}) = (𝑦𝑤 ↦ {𝑥, 𝑦}))
3633, 35eqtrid 2788 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑦𝑤 ↦ {𝑥, 𝑦}))
3736rneqd 5893 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑦𝑤 ↦ {𝑥, 𝑦}))
3831, 37uneq12d 4124 . . . . . . . . . . . . . 14 (𝑢 = 𝑥 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦})))
3938cbviunv 5000 . . . . . . . . . . . . 13 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦}))
40 mpteq1 5198 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → (𝑦𝑤 ↦ {𝑥, 𝑦}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
4140rneqd 5893 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → ran (𝑦𝑤 ↦ {𝑥, 𝑦}) = ran (𝑦𝑧 ↦ {𝑥, 𝑦}))
4241uneq2d 4123 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 → ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
4326, 42iuneq12d 4982 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
4439, 43eqtrid 2788 . . . . . . . . . . . 12 (𝑤 = 𝑧 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
4528, 44uneq12d 4124 . . . . . . . . . . 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 4876 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑚) → 𝑤 = (𝐹𝑚))
4846, 47uneq12d 4124 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑚) → (𝑤 𝑤) = ((𝐹𝑚) ∪ (𝐹𝑚)))
49 mpteq1 5198 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝑚) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))
5049rneqd 5893 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑚) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))
5150uneq2d 4123 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑚) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
5246, 51iuneq12d 4982 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑚) → 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
5348, 52uneq12d 4124 . . . . . . . . . . 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 8386 . . . . . . . . . 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 3985 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 ⊆ (𝐹‘suc 𝑚))
57 fvssunirn 6875 . . . . . . . . 9 (𝐹‘suc 𝑚) ⊆ ran 𝐹
5857, 1sseqtrri 3981 . . . . . . . 8 (𝐹‘suc 𝑚) ⊆ 𝑈
5956, 58sstrdi 3956 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎𝑈)
6059rexlimdvaa 3153 . . . . . 6 (𝐴𝑉 → (∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚) → 𝑎𝑈))
619, 60biimtrid 241 . . . . 5 (𝐴𝑉 → (𝑎𝑈𝑎𝑈))
6261ralrimiv 3142 . . . 4 (𝐴𝑉 → ∀𝑎𝑈 𝑎𝑈)
63 dftr3 5228 . . . 4 (Tr 𝑈 ↔ ∀𝑎𝑈 𝑎𝑈)
6462, 63sylibr 233 . . 3 (𝐴𝑉 → Tr 𝑈)
65 1on 8424 . . . . . . . 8 1o ∈ On
66 unexg 7683 . . . . . . . 8 ((𝐴𝑉 ∧ 1o ∈ On) → (𝐴 ∪ 1o) ∈ V)
6765, 66mpan2 689 . . . . . . 7 (𝐴𝑉 → (𝐴 ∪ 1o) ∈ V)
684fveq1i 6843 . . . . . . . 8 (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅)
69 fr0g 8382 . . . . . . . 8 ((𝐴 ∪ 1o) ∈ V → ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅) = (𝐴 ∪ 1o))
7068, 69eqtrid 2788 . . . . . . 7 ((𝐴 ∪ 1o) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1o))
7167, 70syl 17 . . . . . 6 (𝐴𝑉 → (𝐹‘∅) = (𝐴 ∪ 1o))
72 fvssunirn 6875 . . . . . . 7 (𝐹‘∅) ⊆ ran 𝐹
7372, 1sseqtrri 3981 . . . . . 6 (𝐹‘∅) ⊆ 𝑈
7471, 73eqsstrrdi 3999 . . . . 5 (𝐴𝑉 → (𝐴 ∪ 1o) ⊆ 𝑈)
7574unssbd 4148 . . . 4 (𝐴𝑉 → 1o𝑈)
76 1n0 8434 . . . 4 1o ≠ ∅
77 ssn0 4360 . . . 4 ((1o𝑈 ∧ 1o ≠ ∅) → 𝑈 ≠ ∅)
7875, 76, 77sylancl 586 . . 3 (𝐴𝑉𝑈 ≠ ∅)
79 pweq 4574 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → 𝒫 𝑢 = 𝒫 𝑎)
80 unieq 4876 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 𝑢 = 𝑎)
8179, 80preq12d 4702 . . . . . . . . . . . . . 14 (𝑢 = 𝑎 → {𝒫 𝑢, 𝑢} = {𝒫 𝑎, 𝑎})
82 preq1 4694 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑎 → {𝑢, 𝑣} = {𝑎, 𝑣})
8382mpteq2dv 5207 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣}))
8483rneqd 5893 . . . . . . . . . . . . . 14 (𝑢 = 𝑎 → ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣}))
8581, 84uneq12d 4124 . . . . . . . . . . . . 13 (𝑢 = 𝑎 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) = ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣})))
8685ssiun2s 5008 . . . . . . . . . . . 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 4133 . . . . . . . . . . . . 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 3997 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ⊆ (𝐹‘suc 𝑚))
9089, 58sstrdi 3956 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ⊆ 𝑈)
9187, 90sstrd 3954 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣})) ⊆ 𝑈)
9291unssad 4147 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → {𝒫 𝑎, 𝑎} ⊆ 𝑈)
93 vpwex 5332 . . . . . . . . . 10 𝒫 𝑎 ∈ V
94 vuniex 7676 . . . . . . . . . 10 𝑎 ∈ V
9593, 94prss 4780 . . . . . . . . 9 ((𝒫 𝑎𝑈 𝑎𝑈) ↔ {𝒫 𝑎, 𝑎} ⊆ 𝑈)
9692, 95sylibr 233 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (𝒫 𝑎𝑈 𝑎𝑈))
9796simprd 496 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎𝑈)
9896simpld 495 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝒫 𝑎𝑈)
991eleq2i 2829 . . . . . . . . . 10 (𝑏𝑈𝑏 ran 𝐹)
100 fnunirn 7201 . . . . . . . . . . 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 7812 . . . . . . . . . . . . . . . . 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 7762 . . . . . . . . . . . . . . . . 17 ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
107103, 104, 105, 106mp3an2i 1466 . . . . . . . . . . . . . . . 16 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → (𝑚𝑛) ∈ ω)
108 ssun1 4132 . . . . . . . . . . . . . . . . 17 𝑚 ⊆ (𝑚𝑛)
109 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
110109sseq2d 3976 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹𝑚)))
111 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑖 → (𝐹𝑘) = (𝐹𝑖))
112111sseq2d 3976 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑖 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹𝑖)))
113 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑘 = suc 𝑖 → (𝐹𝑘) = (𝐹‘suc 𝑖))
114113sseq2d 3976 . . . . . . . . . . . . . . . . . 18 (𝑘 = suc 𝑖 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹‘suc 𝑖)))
115 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑚𝑛) → (𝐹𝑘) = (𝐹‘(𝑚𝑛)))
116115sseq2d 3976 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑚𝑛) → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛))))
117 ssidd 3967 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ω → (𝐹𝑚) ⊆ (𝐹𝑚))
118 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑖 → (𝐹𝑚) = (𝐹𝑖))
119 suceq 6383 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑖 → suc 𝑚 = suc 𝑖)
120119fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑖 → (𝐹‘suc 𝑚) = (𝐹‘suc 𝑖))
121118, 120sseq12d 3977 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑖 → ((𝐹𝑚) ⊆ (𝐹‘suc 𝑚) ↔ (𝐹𝑖) ⊆ (𝐹‘suc 𝑖)))
122 ssun1 4132 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹𝑚) ⊆ ((𝐹𝑚) ∪ (𝐹𝑚))
123122, 13sstri 3953 . . . . . . . . . . . . . . . . . . . . . 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 3997 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ω → (𝐹𝑚) ⊆ (𝐹‘suc 𝑚))
126121, 125vtoclga 3534 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ω → (𝐹𝑖) ⊆ (𝐹‘suc 𝑖))
127126ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑖) ⊆ (𝐹‘suc 𝑖))
128 sstr2 3951 . . . . . . . . . . . . . . . . . . 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 7836 . . . . . . . . . . . . . . . . 17 ((((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚 ⊆ (𝑚𝑛)) → (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛)))
131108, 130mpan2 689 . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑎 ∈ (𝐹𝑚))
134132, 133sseldd 3945 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑎 ∈ (𝐹‘(𝑚𝑛)))
13582mpteq2dv 5207 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑎 → (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
136135rneqd 5893 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑎 → ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
13781, 136uneq12d 4124 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) = ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})))
138137ssiun2s 5008 . . . . . . . . . . . . . 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 4133 . . . . . . . . . . . . . . 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 6855 . . . . . . . . . . . . . . . . . 18 (𝐹‘(𝑚𝑛)) ∈ V
142141uniex 7678 . . . . . . . . . . . . . . . . . 18 (𝐹‘(𝑚𝑛)) ∈ V
143141, 142unex 7680 . . . . . . . . . . . . . . . . 17 ((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))) ∈ V
144141mptex 7173 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) ∈ V
145144rnex 7849 . . . . . . . . . . . . . . . . . . 19 ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) ∈ V
14620, 145unex 7680 . . . . . . . . . . . . . . . . . 18 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ∈ V
147141, 146iunex 7901 . . . . . . . . . . . . . . . . 17 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ∈ V
148143, 147unex 7680 . . . . . . . . . . . . . . . 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 4876 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹‘(𝑚𝑛)) → 𝑤 = (𝐹‘(𝑚𝑛)))
151149, 150uneq12d 4124 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝐹‘(𝑚𝑛)) → (𝑤 𝑤) = ((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))))
152 mpteq1 5198 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐹‘(𝑚𝑛)) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))
153152rneqd 5893 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐹‘(𝑚𝑛)) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))
154153uneq2d 4123 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹‘(𝑚𝑛)) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})))
155149, 154iuneq12d 4982 . . . . . . . . . . . . . . . . . 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 4124 . . . . . . . . . . . . . . . . 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 8386 . . . . . . . . . . . . . . . 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 3997 . . . . . . . . . . . . . 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 6875 . . . . . . . . . . . . . . 15 (𝐹‘suc (𝑚𝑛)) ⊆ ran 𝐹
161160, 1sseqtrri 3981 . . . . . . . . . . . . . 14 (𝐹‘suc (𝑚𝑛)) ⊆ 𝑈
162159, 161sstrdi 3956 . . . . . . . . . . . . 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 3954 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})) ⊆ 𝑈)
164163unssbd 4148 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}) ⊆ 𝑈)
165 ssun2 4133 . . . . . . . . . . . . . . . . . . 19 𝑛 ⊆ (𝑚𝑛)
166 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑖 = (𝑚𝑛) → 𝑖 = (𝑚𝑛))
167165, 166sseqtrrid 3997 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑚𝑛) → 𝑛𝑖)
168167biantrud 532 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚𝑛) → (𝑛 ∈ ω ↔ (𝑛 ∈ ω ∧ 𝑛𝑖)))
169168bicomd 222 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚𝑛) → ((𝑛 ∈ ω ∧ 𝑛𝑖) ↔ 𝑛 ∈ ω))
170 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚𝑛) → (𝐹𝑖) = (𝐹‘(𝑚𝑛)))
171170sseq2d 3976 . . . . . . . . . . . . . . . 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 2820 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑚 ∈ ω ↔ 𝑛 ∈ ω))
174173anbi2d 629 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → ((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ↔ (𝑖 ∈ ω ∧ 𝑛 ∈ ω)))
175 sseq1 3969 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → (𝑚𝑖𝑛𝑖))
176174, 175anbi12d 631 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) ↔ ((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖)))
177 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
178177sseq1d 3975 . . . . . . . . . . . . . . . . . 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 7836 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑚) ⊆ (𝐹𝑖))
181179, 180chvarvv 2002 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖))
182181expl 458 . . . . . . . . . . . . . . 15 (𝑖 ∈ ω → ((𝑛 ∈ ω ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖)))
183172, 182vtoclga 3534 . . . . . . . . . . . . . 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 3945 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑏 ∈ (𝐹‘(𝑚𝑛)))
187 prex 5389 . . . . . . . . . . . 12 {𝑎, 𝑏} ∈ V
188 eqid 2736 . . . . . . . . . . . . 13 (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})
189 preq2 4695 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → {𝑎, 𝑣} = {𝑎, 𝑏})
190188, 189elrnmpt1s 5912 . . . . . . . . . . . 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 3945 . . . . . . . . . 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 241 . . . . . . . 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 1128 . . . . . 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 241 . . . 4 (𝐴𝑉 → (𝑎𝑈 → ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)))
199198ralrimiv 3142 . . 3 (𝐴𝑉 → ∀𝑎𝑈 ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈))
200 rdgfun 8362 . . . . . . . . 9 Fun rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o))
201 omex 9579 . . . . . . . . 9 ω ∈ V
202 resfunexg 7165 . . . . . . . . 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 2834 . . . . . . 7 𝐹 ∈ V
205204rnex 7849 . . . . . 6 ran 𝐹 ∈ V
206205uniex 7678 . . . . 5 ran 𝐹 ∈ V
2071, 206eqeltri 2834 . . . 4 𝑈 ∈ V
208 iswun 10640 . . . 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 1343 . 2 (𝐴𝑉𝑈 ∈ WUni)
21174unssad 4147 . 2 (𝐴𝑉𝐴𝑈)
212210, 211jca 512 1 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cun 3908  wss 3910  c0 4282  𝒫 cpw 4560  {cpr 4588   cuni 4865   ciun 4954  cmpt 5188  Tr wtr 5222  ran crn 5634  cres 5635  Ord word 6316  Oncon0 6317  suc csuc 6319  Fun wfun 6490   Fn wfn 6491  cfv 6496  ωcom 7802  reccrdg 8355  1oc1o 8405  WUnicwun 10636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-wun 10638
This theorem is referenced by:  wunex  10675  wuncval2  10683
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