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Theorem wuncval2 10785
Description: Our earlier expression for a containing weak universe is in fact the weak universe closure. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
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
Ref Expression
wuncval2 (𝐴𝑉 → (wUniCl‘𝐴) = 𝑈)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   𝐴(𝑧)   𝑈(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)   𝑉(𝑧)

Proof of Theorem wuncval2
Dummy variables 𝑣 𝑢 𝑤 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wuncval2.f . . . 4 𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)
2 wuncval2.u . . . 4 𝑈 = ran 𝐹
31, 2wunex2 10776 . . 3 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
4 wuncss 10783 . . 3 ((𝑈 ∈ WUni ∧ 𝐴𝑈) → (wUniCl‘𝐴) ⊆ 𝑈)
53, 4syl 17 . 2 (𝐴𝑉 → (wUniCl‘𝐴) ⊆ 𝑈)
6 frfnom 8474 . . . . . 6 (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) Fn ω
71fneq1i 6666 . . . . . 6 (𝐹 Fn ω ↔ (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) Fn ω)
86, 7mpbir 231 . . . . 5 𝐹 Fn ω
9 fniunfv 7267 . . . . 5 (𝐹 Fn ω → 𝑚 ∈ ω (𝐹𝑚) = ran 𝐹)
108, 9ax-mp 5 . . . 4 𝑚 ∈ ω (𝐹𝑚) = ran 𝐹
112, 10eqtr4i 2766 . . 3 𝑈 = 𝑚 ∈ ω (𝐹𝑚)
12 fveq2 6907 . . . . . . . 8 (𝑚 = ∅ → (𝐹𝑚) = (𝐹‘∅))
1312sseq1d 4027 . . . . . . 7 (𝑚 = ∅ → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘∅) ⊆ (wUniCl‘𝐴)))
14 fveq2 6907 . . . . . . . 8 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
1514sseq1d 4027 . . . . . . 7 (𝑚 = 𝑛 → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹𝑛) ⊆ (wUniCl‘𝐴)))
16 fveq2 6907 . . . . . . . 8 (𝑚 = suc 𝑛 → (𝐹𝑚) = (𝐹‘suc 𝑛))
1716sseq1d 4027 . . . . . . 7 (𝑚 = suc 𝑛 → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
18 1on 8517 . . . . . . . . . 10 1o ∈ On
19 unexg 7762 . . . . . . . . . 10 ((𝐴𝑉 ∧ 1o ∈ On) → (𝐴 ∪ 1o) ∈ V)
2018, 19mpan2 691 . . . . . . . . 9 (𝐴𝑉 → (𝐴 ∪ 1o) ∈ V)
211fveq1i 6908 . . . . . . . . . 10 (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅)
22 fr0g 8475 . . . . . . . . . 10 ((𝐴 ∪ 1o) ∈ V → ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅) = (𝐴 ∪ 1o))
2321, 22eqtrid 2787 . . . . . . . . 9 ((𝐴 ∪ 1o) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1o))
2420, 23syl 17 . . . . . . . 8 (𝐴𝑉 → (𝐹‘∅) = (𝐴 ∪ 1o))
25 wuncid 10781 . . . . . . . . 9 (𝐴𝑉𝐴 ⊆ (wUniCl‘𝐴))
26 df1o2 8512 . . . . . . . . . 10 1o = {∅}
27 wunccl 10782 . . . . . . . . . . . 12 (𝐴𝑉 → (wUniCl‘𝐴) ∈ WUni)
2827wun0 10756 . . . . . . . . . . 11 (𝐴𝑉 → ∅ ∈ (wUniCl‘𝐴))
2928snssd 4814 . . . . . . . . . 10 (𝐴𝑉 → {∅} ⊆ (wUniCl‘𝐴))
3026, 29eqsstrid 4044 . . . . . . . . 9 (𝐴𝑉 → 1o ⊆ (wUniCl‘𝐴))
3125, 30unssd 4202 . . . . . . . 8 (𝐴𝑉 → (𝐴 ∪ 1o) ⊆ (wUniCl‘𝐴))
3224, 31eqsstrd 4034 . . . . . . 7 (𝐴𝑉 → (𝐹‘∅) ⊆ (wUniCl‘𝐴))
33 simplr 769 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → 𝑛 ∈ ω)
34 fvex 6920 . . . . . . . . . . . . 13 (𝐹𝑛) ∈ V
3534uniex 7760 . . . . . . . . . . . . 13 (𝐹𝑛) ∈ V
3634, 35unex 7763 . . . . . . . . . . . 12 ((𝐹𝑛) ∪ (𝐹𝑛)) ∈ V
37 prex 5443 . . . . . . . . . . . . . 14 {𝒫 𝑢, 𝑢} ∈ V
3834mptex 7243 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ∈ V
3938rnex 7933 . . . . . . . . . . . . . 14 ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ∈ V
4037, 39unex 7763 . . . . . . . . . . . . 13 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ∈ V
4134, 40iunex 7992 . . . . . . . . . . . 12 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ∈ V
4236, 41unex 7763 . . . . . . . . . . 11 (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))) ∈ V
43 id 22 . . . . . . . . . . . . . 14 (𝑤 = 𝑧𝑤 = 𝑧)
44 unieq 4923 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 𝑤 = 𝑧)
4543, 44uneq12d 4179 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑤 𝑤) = (𝑧 𝑧))
46 pweq 4619 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
47 unieq 4923 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 𝑢 = 𝑥)
4846, 47preq12d 4746 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → {𝒫 𝑢, 𝑢} = {𝒫 𝑥, 𝑥})
49 preq1 4738 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑥 → {𝑢, 𝑣} = {𝑥, 𝑣})
5049mpteq2dv 5250 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣𝑤 ↦ {𝑥, 𝑣}))
5150rneqd 5952 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣𝑤 ↦ {𝑥, 𝑣}))
5248, 51uneq12d 4179 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})))
5352cbviunv 5045 . . . . . . . . . . . . . 14 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣}))
54 preq2 4739 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑦 → {𝑥, 𝑣} = {𝑥, 𝑦})
5554cbvmptv 5261 . . . . . . . . . . . . . . . . . 18 (𝑣𝑤 ↦ {𝑥, 𝑣}) = (𝑦𝑤 ↦ {𝑥, 𝑦})
56 mpteq1 5241 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑦𝑤 ↦ {𝑥, 𝑦}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
5755, 56eqtrid 2787 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑣𝑤 ↦ {𝑥, 𝑣}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
5857rneqd 5952 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → ran (𝑣𝑤 ↦ {𝑥, 𝑣}) = ran (𝑦𝑧 ↦ {𝑥, 𝑦}))
5958uneq2d 4178 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6043, 59iuneq12d 5026 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6153, 60eqtrid 2787 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6245, 61uneq12d 4179 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ((𝑤 𝑤) ∪ 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣}))) = ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦}))))
63 id 22 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑛) → 𝑤 = (𝐹𝑛))
64 unieq 4923 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑛) → 𝑤 = (𝐹𝑛))
6563, 64uneq12d 4179 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑛) → (𝑤 𝑤) = ((𝐹𝑛) ∪ (𝐹𝑛)))
66 mpteq1 5241 . . . . . . . . . . . . . . . 16 (𝑤 = (𝐹𝑛) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))
6766rneqd 5952 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝑛) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))
6867uneq2d 4178 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑛) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})))
6963, 68iuneq12d 5026 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑛) → 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})))
7065, 69uneq12d 4179 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑛) → ((𝑤 𝑤) ∪ 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣}))) = (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))))
711, 62, 70frsucmpt2 8479 . . . . . . . . . . 11 ((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}))))
7233, 42, 71sylancl 586 . . . . . . . . . 10 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘suc 𝑛) = (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))))
73 simpr 484 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
7427ad3antrrr 730 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (wUniCl‘𝐴) ∈ WUni)
7573sselda 3995 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
7674, 75wunelss 10746 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ⊆ (wUniCl‘𝐴))
7776ralrimiva 3144 . . . . . . . . . . . . 13 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹𝑛)𝑢 ⊆ (wUniCl‘𝐴))
78 unissb 4944 . . . . . . . . . . . . 13 ( (𝐹𝑛) ⊆ (wUniCl‘𝐴) ↔ ∀𝑢 ∈ (𝐹𝑛)𝑢 ⊆ (wUniCl‘𝐴))
7977, 78sylibr 234 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
8073, 79unssd 4202 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ((𝐹𝑛) ∪ (𝐹𝑛)) ⊆ (wUniCl‘𝐴))
8174, 75wunpw 10745 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝒫 𝑢 ∈ (wUniCl‘𝐴))
8274, 75wununi 10744 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
8381, 82prssd 4827 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → {𝒫 𝑢, 𝑢} ⊆ (wUniCl‘𝐴))
8474adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → (wUniCl‘𝐴) ∈ WUni)
8575adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
86 simplr 769 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
8786sselda 3995 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → 𝑣 ∈ (wUniCl‘𝐴))
8884, 85, 87wunpr 10747 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → {𝑢, 𝑣} ∈ (wUniCl‘𝐴))
8988fmpttd 7135 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}):(𝐹𝑛)⟶(wUniCl‘𝐴))
9089frnd 6745 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ⊆ (wUniCl‘𝐴))
9183, 90unssd 4202 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
9291ralrimiva 3144 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
93 iunss 5050 . . . . . . . . . . . 12 ( 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴) ↔ ∀𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
9492, 93sylibr 234 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
9580, 94unssd 4202 . . . . . . . . . 10 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))) ⊆ (wUniCl‘𝐴))
9672, 95eqsstrd 4034 . . . . . . . . 9 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))
9796ex 412 . . . . . . . 8 ((𝐴𝑉𝑛 ∈ ω) → ((𝐹𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
9897expcom 413 . . . . . . 7 (𝑛 ∈ ω → (𝐴𝑉 → ((𝐹𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))))
9913, 15, 17, 32, 98finds2 7921 . . . . . 6 (𝑚 ∈ ω → (𝐴𝑉 → (𝐹𝑚) ⊆ (wUniCl‘𝐴)))
10099com12 32 . . . . 5 (𝐴𝑉 → (𝑚 ∈ ω → (𝐹𝑚) ⊆ (wUniCl‘𝐴)))
101100ralrimiv 3143 . . . 4 (𝐴𝑉 → ∀𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
102 iunss 5050 . . . 4 ( 𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ ∀𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
103101, 102sylibr 234 . . 3 (𝐴𝑉 𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
10411, 103eqsstrid 4044 . 2 (𝐴𝑉𝑈 ⊆ (wUniCl‘𝐴))
1055, 104eqssd 4013 1 (𝐴𝑉 → (wUniCl‘𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cun 3961  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631  {cpr 4633   cuni 4912   ciun 4996  cmpt 5231  ran crn 5690  cres 5691  Oncon0 6386  suc csuc 6388   Fn wfn 6558  cfv 6563  ωcom 7887  reccrdg 8448  1oc1o 8498  WUnicwun 10738  wUniClcwunm 10739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-wun 10740  df-wunc 10741
This theorem is referenced by: (None)
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