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Theorem wunex2 9952
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 2851 . . . . . . 7 (𝑎𝑈𝑎 ran 𝐹)
3 frfnom 7868 . . . . . . . . 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 6277 . . . . . . . . 9 (𝐹 Fn ω ↔ (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) Fn ω)
63, 5mpbir 223 . . . . . . . 8 𝐹 Fn ω
7 fnunirn 6831 . . . . . . . 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 267 . . . . . 6 (𝑎𝑈 ↔ ∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚))
10 elssuni 4735 . . . . . . . . . . 11 (𝑎 ∈ (𝐹𝑚) → 𝑎 (𝐹𝑚))
1110ad2antll 716 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 (𝐹𝑚))
12 ssun2 4032 . . . . . . . . . . 11 (𝐹𝑚) ⊆ ((𝐹𝑚) ∪ (𝐹𝑚))
13 ssun1 4031 . . . . . . . . . . 11 ((𝐹𝑚) ∪ (𝐹𝑚)) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
1412, 13sstri 3861 . . . . . . . . . 10 (𝐹𝑚) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
1511, 14syl6ss 3864 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
16 simprl 758 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑚 ∈ ω)
17 fvex 6506 . . . . . . . . . . . 12 (𝐹𝑚) ∈ V
1817uniex 7277 . . . . . . . . . . . 12 (𝐹𝑚) ∈ V
1917, 18unex 7280 . . . . . . . . . . 11 ((𝐹𝑚) ∪ (𝐹𝑚)) ∈ V
20 prex 5183 . . . . . . . . . . . . 13 {𝒫 𝑢, 𝑢} ∈ V
2117mptex 6806 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) ∈ V
2221rnex 7426 . . . . . . . . . . . . 13 ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) ∈ V
2320, 22unex 7280 . . . . . . . . . . . 12 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ∈ V
2417, 23iunex 7475 . . . . . . . . . . 11 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ∈ V
2519, 24unex 7280 . . . . . . . . . 10 (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))) ∈ V
26 id 22 . . . . . . . . . . . . 13 (𝑤 = 𝑧𝑤 = 𝑧)
27 unieq 4714 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑤 = 𝑧)
2826, 27uneq12d 4023 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑤 𝑤) = (𝑧 𝑧))
29 pweq 4419 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
30 unieq 4714 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 𝑢 = 𝑥)
3129, 30preq12d 4545 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → {𝒫 𝑢, 𝑢} = {𝒫 𝑥, 𝑥})
32 preq2 4538 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑦 → {𝑢, 𝑣} = {𝑢, 𝑦})
3332cbvmptv 5022 . . . . . . . . . . . . . . . . 17 (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑦𝑤 ↦ {𝑢, 𝑦})
34 preq1 4537 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑥 → {𝑢, 𝑦} = {𝑥, 𝑦})
3534mpteq2dv 5017 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → (𝑦𝑤 ↦ {𝑢, 𝑦}) = (𝑦𝑤 ↦ {𝑥, 𝑦}))
3633, 35syl5eq 2820 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑦𝑤 ↦ {𝑥, 𝑦}))
3736rneqd 5645 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑦𝑤 ↦ {𝑥, 𝑦}))
3831, 37uneq12d 4023 . . . . . . . . . . . . . 14 (𝑢 = 𝑥 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦})))
3938cbviunv 4827 . . . . . . . . . . . . 13 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦}))
40 mpteq1 5009 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → (𝑦𝑤 ↦ {𝑥, 𝑦}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
4140rneqd 5645 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → ran (𝑦𝑤 ↦ {𝑥, 𝑦}) = ran (𝑦𝑧 ↦ {𝑥, 𝑦}))
4241uneq2d 4022 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 → ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
4326, 42iuneq12d 4813 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑤 ↦ {𝑥, 𝑦})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
4439, 43syl5eq 2820 . . . . . . . . . . . 12 (𝑤 = 𝑧 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
4528, 44uneq12d 4023 . . . . . . . . . . 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 4714 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑚) → 𝑤 = (𝐹𝑚))
4846, 47uneq12d 4023 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑚) → (𝑤 𝑤) = ((𝐹𝑚) ∪ (𝐹𝑚)))
49 mpteq1 5009 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝑚) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))
5049rneqd 5645 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑚) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))
5150uneq2d 4022 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑚) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
5246, 51iuneq12d 4813 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑚) → 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
5348, 52uneq12d 4023 . . . . . . . . . . 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 7873 . . . . . . . . . 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 577 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (𝐹‘suc 𝑚) = (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
5615, 55sseqtr4d 3892 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎 ⊆ (𝐹‘suc 𝑚))
57 fvssunirn 6522 . . . . . . . . 9 (𝐹‘suc 𝑚) ⊆ ran 𝐹
5857, 1sseqtr4i 3888 . . . . . . . 8 (𝐹‘suc 𝑚) ⊆ 𝑈
5956, 58syl6ss 3864 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎𝑈)
6059rexlimdvaa 3224 . . . . . 6 (𝐴𝑉 → (∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚) → 𝑎𝑈))
619, 60syl5bi 234 . . . . 5 (𝐴𝑉 → (𝑎𝑈𝑎𝑈))
6261ralrimiv 3125 . . . 4 (𝐴𝑉 → ∀𝑎𝑈 𝑎𝑈)
63 dftr3 5028 . . . 4 (Tr 𝑈 ↔ ∀𝑎𝑈 𝑎𝑈)
6462, 63sylibr 226 . . 3 (𝐴𝑉 → Tr 𝑈)
65 1on 7906 . . . . . . . 8 1o ∈ On
66 unexg 7283 . . . . . . . 8 ((𝐴𝑉 ∧ 1o ∈ On) → (𝐴 ∪ 1o) ∈ V)
6765, 66mpan2 678 . . . . . . 7 (𝐴𝑉 → (𝐴 ∪ 1o) ∈ V)
684fveq1i 6494 . . . . . . . 8 (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅)
69 fr0g 7869 . . . . . . . 8 ((𝐴 ∪ 1o) ∈ V → ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅) = (𝐴 ∪ 1o))
7068, 69syl5eq 2820 . . . . . . 7 ((𝐴 ∪ 1o) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1o))
7167, 70syl 17 . . . . . 6 (𝐴𝑉 → (𝐹‘∅) = (𝐴 ∪ 1o))
72 fvssunirn 6522 . . . . . . 7 (𝐹‘∅) ⊆ ran 𝐹
7372, 1sseqtr4i 3888 . . . . . 6 (𝐹‘∅) ⊆ 𝑈
7471, 73syl6eqssr 3906 . . . . 5 (𝐴𝑉 → (𝐴 ∪ 1o) ⊆ 𝑈)
7574unssbd 4046 . . . 4 (𝐴𝑉 → 1o𝑈)
76 1n0 7915 . . . 4 1o ≠ ∅
77 ssn0 4234 . . . 4 ((1o𝑈 ∧ 1o ≠ ∅) → 𝑈 ≠ ∅)
7875, 76, 77sylancl 577 . . 3 (𝐴𝑉𝑈 ≠ ∅)
79 pweq 4419 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → 𝒫 𝑢 = 𝒫 𝑎)
80 unieq 4714 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 𝑢 = 𝑎)
8179, 80preq12d 4545 . . . . . . . . . . . . . 14 (𝑢 = 𝑎 → {𝒫 𝑢, 𝑢} = {𝒫 𝑎, 𝑎})
82 preq1 4537 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑎 → {𝑢, 𝑣} = {𝑎, 𝑣})
8382mpteq2dv 5017 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣}))
8483rneqd 5645 . . . . . . . . . . . . . 14 (𝑢 = 𝑎 → ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣}))
8581, 84uneq12d 4023 . . . . . . . . . . . . 13 (𝑢 = 𝑎 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) = ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣})))
8685ssiun2s 4832 . . . . . . . . . . . 12 (𝑎 ∈ (𝐹𝑚) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣})) ⊆ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
8786ad2antll 716 . . . . . . . . . . 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 4032 . . . . . . . . . . . . 13 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
8988, 55syl5sseqr 3904 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ⊆ (𝐹‘suc 𝑚))
9089, 58syl6ss 3864 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})) ⊆ 𝑈)
9187, 90sstrd 3862 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑎, 𝑣})) ⊆ 𝑈)
9291unssad 4045 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → {𝒫 𝑎, 𝑎} ⊆ 𝑈)
93 vpwex 5125 . . . . . . . . . 10 𝒫 𝑎 ∈ V
94 vuniex 7278 . . . . . . . . . 10 𝑎 ∈ V
9593, 94prss 4621 . . . . . . . . 9 ((𝒫 𝑎𝑈 𝑎𝑈) ↔ {𝒫 𝑎, 𝑎} ⊆ 𝑈)
9692, 95sylibr 226 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (𝒫 𝑎𝑈 𝑎𝑈))
9796simprd 488 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝑎𝑈)
9896simpld 487 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → 𝒫 𝑎𝑈)
991eleq2i 2851 . . . . . . . . . 10 (𝑏𝑈𝑏 ran 𝐹)
100 fnunirn 6831 . . . . . . . . . . 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 267 . . . . . . . . 9 (𝑏𝑈 ↔ ∃𝑛 ∈ ω 𝑏 ∈ (𝐹𝑛))
103 ordom 7399 . . . . . . . . . . . . . . . . 17 Ord ω
104 simplrl 764 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑚 ∈ ω)
105 simprl 758 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑛 ∈ ω)
106 ordunel 7352 . . . . . . . . . . . . . . . . 17 ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
107103, 104, 105, 106mp3an2i 1445 . . . . . . . . . . . . . . . 16 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → (𝑚𝑛) ∈ ω)
108 ssun1 4031 . . . . . . . . . . . . . . . . 17 𝑚 ⊆ (𝑚𝑛)
109 fveq2 6493 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
110109sseq2d 3883 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹𝑚)))
111 fveq2 6493 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑖 → (𝐹𝑘) = (𝐹𝑖))
112111sseq2d 3883 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑖 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹𝑖)))
113 fveq2 6493 . . . . . . . . . . . . . . . . . . 19 (𝑘 = suc 𝑖 → (𝐹𝑘) = (𝐹‘suc 𝑖))
114113sseq2d 3883 . . . . . . . . . . . . . . . . . 18 (𝑘 = suc 𝑖 → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹‘suc 𝑖)))
115 fveq2 6493 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑚𝑛) → (𝐹𝑘) = (𝐹‘(𝑚𝑛)))
116115sseq2d 3883 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑚𝑛) → ((𝐹𝑚) ⊆ (𝐹𝑘) ↔ (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛))))
117 ssidd 3874 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ω → (𝐹𝑚) ⊆ (𝐹𝑚))
118 fveq2 6493 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑖 → (𝐹𝑚) = (𝐹𝑖))
119 suceq 6088 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑖 → suc 𝑚 = suc 𝑖)
120119fveq2d 6497 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑖 → (𝐹‘suc 𝑚) = (𝐹‘suc 𝑖))
121118, 120sseq12d 3884 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑖 → ((𝐹𝑚) ⊆ (𝐹‘suc 𝑚) ↔ (𝐹𝑖) ⊆ (𝐹‘suc 𝑖)))
122 ssun1 4031 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹𝑚) ⊆ ((𝐹𝑚) ∪ (𝐹𝑚))
123122, 13sstri 3861 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹𝑚) ⊆ (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣})))
12425, 54mpan2 678 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ω → (𝐹‘suc 𝑚) = (((𝐹𝑚) ∪ (𝐹𝑚)) ∪ 𝑢 ∈ (𝐹𝑚)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑚) ↦ {𝑢, 𝑣}))))
125123, 124syl5sseqr 3904 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ω → (𝐹𝑚) ⊆ (𝐹‘suc 𝑚))
126121, 125vtoclga 3487 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ω → (𝐹𝑖) ⊆ (𝐹‘suc 𝑖))
127126ad2antrr 713 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑖) ⊆ (𝐹‘suc 𝑖))
128 sstr2 3859 . . . . . . . . . . . . . . . . . . 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 7418 . . . . . . . . . . . . . . . . 17 ((((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚 ⊆ (𝑚𝑛)) → (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛)))
131108, 130mpan2 678 . . . . . . . . . . . . . . . 16 (((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω) → (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛)))
132107, 104, 131syl2anc 576 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → (𝐹𝑚) ⊆ (𝐹‘(𝑚𝑛)))
133 simplrr 765 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑎 ∈ (𝐹𝑚))
134132, 133sseldd 3853 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑎 ∈ (𝐹‘(𝑚𝑛)))
13582mpteq2dv 5017 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑎 → (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
136135rneqd 5645 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑎 → ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
13781, 136uneq12d 4023 . . . . . . . . . . . . . . 15 (𝑢 = 𝑎 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) = ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})))
138137ssiun2s 4832 . . . . . . . . . . . . . 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 4032 . . . . . . . . . . . . . . 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 6506 . . . . . . . . . . . . . . . . . 18 (𝐹‘(𝑚𝑛)) ∈ V
142141uniex 7277 . . . . . . . . . . . . . . . . . 18 (𝐹‘(𝑚𝑛)) ∈ V
143141, 142unex 7280 . . . . . . . . . . . . . . . . 17 ((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))) ∈ V
144141mptex 6806 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) ∈ V
145144rnex 7426 . . . . . . . . . . . . . . . . . . 19 ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}) ∈ V
14620, 145unex 7280 . . . . . . . . . . . . . . . . . 18 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ∈ V
147141, 146iunex 7475 . . . . . . . . . . . . . . . . 17 𝑢 ∈ (𝐹‘(𝑚𝑛))({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})) ∈ V
148143, 147unex 7280 . . . . . . . . . . . . . . . 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 4714 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹‘(𝑚𝑛)) → 𝑤 = (𝐹‘(𝑚𝑛)))
151149, 150uneq12d 4023 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝐹‘(𝑚𝑛)) → (𝑤 𝑤) = ((𝐹‘(𝑚𝑛)) ∪ (𝐹‘(𝑚𝑛))))
152 mpteq1 5009 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐹‘(𝑚𝑛)) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))
153152rneqd 5645 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐹‘(𝑚𝑛)) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣}))
154153uneq2d 4022 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹‘(𝑚𝑛)) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑢, 𝑣})))
155149, 154iuneq12d 4813 . . . . . . . . . . . . . . . . . 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 4023 . . . . . . . . . . . . . . . . 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 7873 . . . . . . . . . . . . . . . 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 577 . . . . . . . . . . . . . . 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, 158syl5sseqr 3904 . . . . . . . . . . . . . 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 6522 . . . . . . . . . . . . . . 15 (𝐹‘suc (𝑚𝑛)) ⊆ ran 𝐹
161160, 1sseqtr4i 3888 . . . . . . . . . . . . . 14 (𝐹‘suc (𝑚𝑛)) ⊆ 𝑈
162159, 161syl6ss 3864 . . . . . . . . . . . . 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 3862 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → ({𝒫 𝑎, 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})) ⊆ 𝑈)
164163unssbd 4046 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}) ⊆ 𝑈)
165 ssun2 4032 . . . . . . . . . . . . . . . . . . 19 𝑛 ⊆ (𝑚𝑛)
166 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑖 = (𝑚𝑛) → 𝑖 = (𝑚𝑛))
167165, 166syl5sseqr 3904 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑚𝑛) → 𝑛𝑖)
168167biantrud 524 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚𝑛) → (𝑛 ∈ ω ↔ (𝑛 ∈ ω ∧ 𝑛𝑖)))
169168bicomd 215 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚𝑛) → ((𝑛 ∈ ω ∧ 𝑛𝑖) ↔ 𝑛 ∈ ω))
170 fveq2 6493 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚𝑛) → (𝐹𝑖) = (𝐹‘(𝑚𝑛)))
171170sseq2d 3883 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚𝑛) → ((𝐹𝑛) ⊆ (𝐹𝑖) ↔ (𝐹𝑛) ⊆ (𝐹‘(𝑚𝑛))))
172169, 171imbi12d 337 . . . . . . . . . . . . . . 15 (𝑖 = (𝑚𝑛) → (((𝑛 ∈ ω ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖)) ↔ (𝑛 ∈ ω → (𝐹𝑛) ⊆ (𝐹‘(𝑚𝑛)))))
173 eleq1w 2842 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑚 ∈ ω ↔ 𝑛 ∈ ω))
174173anbi2d 619 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → ((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ↔ (𝑖 ∈ ω ∧ 𝑛 ∈ ω)))
175 sseq1 3876 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → (𝑚𝑖𝑛𝑖))
176174, 175anbi12d 621 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) ↔ ((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖)))
177 fveq2 6493 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
178177sseq1d 3882 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → ((𝐹𝑚) ⊆ (𝐹𝑖) ↔ (𝐹𝑛) ⊆ (𝐹𝑖)))
179176, 178imbi12d 337 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → ((((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑚) ⊆ (𝐹𝑖)) ↔ (((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖))))
180110, 112, 114, 112, 117, 129findsg 7418 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚𝑖) → (𝐹𝑚) ⊆ (𝐹𝑖))
181179, 180chvarv 2327 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖))
182181expl 450 . . . . . . . . . . . . . . 15 (𝑖 ∈ ω → ((𝑛 ∈ ω ∧ 𝑛𝑖) → (𝐹𝑛) ⊆ (𝐹𝑖)))
183172, 182vtoclga 3487 . . . . . . . . . . . . . 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 760 . . . . . . . . . . . . 13 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑏 ∈ (𝐹𝑛))
186184, 185sseldd 3853 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → 𝑏 ∈ (𝐹‘(𝑚𝑛)))
187 prex 5183 . . . . . . . . . . . 12 {𝑎, 𝑏} ∈ V
188 eqid 2772 . . . . . . . . . . . . 13 (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣})
189 preq2 4538 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → {𝑎, 𝑣} = {𝑎, 𝑏})
190188, 189elrnmpt1s 5666 . . . . . . . . . . . 12 ((𝑏 ∈ (𝐹‘(𝑚𝑛)) ∧ {𝑎, 𝑏} ∈ V) → {𝑎, 𝑏} ∈ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
191186, 187, 190sylancl 577 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → {𝑎, 𝑏} ∈ ran (𝑣 ∈ (𝐹‘(𝑚𝑛)) ↦ {𝑎, 𝑣}))
192164, 191sseldd 3853 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹𝑛))) → {𝑎, 𝑏} ∈ 𝑈)
193192rexlimdvaa 3224 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (∃𝑛 ∈ ω 𝑏 ∈ (𝐹𝑛) → {𝑎, 𝑏} ∈ 𝑈))
194102, 193syl5bi 234 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → (𝑏𝑈 → {𝑎, 𝑏} ∈ 𝑈))
195194ralrimiv 3125 . . . . . . 7 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)
19697, 98, 1953jca 1108 . . . . . 6 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹𝑚))) → ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈))
197196rexlimdvaa 3224 . . . . 5 (𝐴𝑉 → (∃𝑚 ∈ ω 𝑎 ∈ (𝐹𝑚) → ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)))
1989, 197syl5bi 234 . . . 4 (𝐴𝑉 → (𝑎𝑈 → ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈)))
199198ralrimiv 3125 . . 3 (𝐴𝑉 → ∀𝑎𝑈 ( 𝑎𝑈 ∧ 𝒫 𝑎𝑈 ∧ ∀𝑏𝑈 {𝑎, 𝑏} ∈ 𝑈))
200 rdgfun 7850 . . . . . . . . 9 Fun rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o))
201 omex 8894 . . . . . . . . 9 ω ∈ V
202 resfunexg 6798 . . . . . . . . 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 679 . . . . . . . 8 (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) ∈ V
2044, 203eqeltri 2856 . . . . . . 7 𝐹 ∈ V
205204rnex 7426 . . . . . 6 ran 𝐹 ∈ V
206205uniex 7277 . . . . 5 ran 𝐹 ∈ V
2071, 206eqeltri 2856 . . . 4 𝑈 ∈ V
208 iswun 9918 . . . 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 1323 . 2 (𝐴𝑉𝑈 ∈ WUni)
21174unssad 4045 . 2 (𝐴𝑉𝐴𝑈)
212210, 211jca 504 1 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2961  wral 3082  wrex 3083  Vcvv 3409  cun 3821  wss 3823  c0 4172  𝒫 cpw 4416  {cpr 4437   cuni 4706   ciun 4786  cmpt 5002  Tr wtr 5024  ran crn 5402  cres 5403  Ord word 6022  Oncon0 6023  suc csuc 6025  Fun wfun 6176   Fn wfn 6177  cfv 6182  ωcom 7390  reccrdg 7843  1oc1o 7892  WUnicwun 9914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-inf2 8892
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-om 7391  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-1o 7899  df-wun 9916
This theorem is referenced by:  wunex  9953  wuncval2  9961
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