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Theorem wuncval2 10146
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 10137 . . 3 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
4 wuncss 10144 . . 3 ((𝑈 ∈ WUni ∧ 𝐴𝑈) → (wUniCl‘𝐴) ⊆ 𝑈)
53, 4syl 17 . 2 (𝐴𝑉 → (wUniCl‘𝐴) ⊆ 𝑈)
6 frfnom 8045 . . . . . 6 (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) Fn ω
71fneq1i 6423 . . . . . 6 (𝐹 Fn ω ↔ (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) Fn ω)
86, 7mpbir 234 . . . . 5 𝐹 Fn ω
9 fniunfv 6980 . . . . 5 (𝐹 Fn ω → 𝑚 ∈ ω (𝐹𝑚) = ran 𝐹)
108, 9ax-mp 5 . . . 4 𝑚 ∈ ω (𝐹𝑚) = ran 𝐹
112, 10eqtr4i 2847 . . 3 𝑈 = 𝑚 ∈ ω (𝐹𝑚)
12 fveq2 6643 . . . . . . . 8 (𝑚 = ∅ → (𝐹𝑚) = (𝐹‘∅))
1312sseq1d 3974 . . . . . . 7 (𝑚 = ∅ → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘∅) ⊆ (wUniCl‘𝐴)))
14 fveq2 6643 . . . . . . . 8 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
1514sseq1d 3974 . . . . . . 7 (𝑚 = 𝑛 → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹𝑛) ⊆ (wUniCl‘𝐴)))
16 fveq2 6643 . . . . . . . 8 (𝑚 = suc 𝑛 → (𝐹𝑚) = (𝐹‘suc 𝑛))
1716sseq1d 3974 . . . . . . 7 (𝑚 = suc 𝑛 → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
18 1on 8084 . . . . . . . . . 10 1o ∈ On
19 unexg 7447 . . . . . . . . . 10 ((𝐴𝑉 ∧ 1o ∈ On) → (𝐴 ∪ 1o) ∈ V)
2018, 19mpan2 690 . . . . . . . . 9 (𝐴𝑉 → (𝐴 ∪ 1o) ∈ V)
211fveq1i 6644 . . . . . . . . . 10 (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅)
22 fr0g 8046 . . . . . . . . . 10 ((𝐴 ∪ 1o) ∈ V → ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅) = (𝐴 ∪ 1o))
2321, 22syl5eq 2868 . . . . . . . . 9 ((𝐴 ∪ 1o) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1o))
2420, 23syl 17 . . . . . . . 8 (𝐴𝑉 → (𝐹‘∅) = (𝐴 ∪ 1o))
25 wuncid 10142 . . . . . . . . 9 (𝐴𝑉𝐴 ⊆ (wUniCl‘𝐴))
26 df1o2 8091 . . . . . . . . . 10 1o = {∅}
27 wunccl 10143 . . . . . . . . . . . 12 (𝐴𝑉 → (wUniCl‘𝐴) ∈ WUni)
2827wun0 10117 . . . . . . . . . . 11 (𝐴𝑉 → ∅ ∈ (wUniCl‘𝐴))
2928snssd 4715 . . . . . . . . . 10 (𝐴𝑉 → {∅} ⊆ (wUniCl‘𝐴))
3026, 29eqsstrid 3991 . . . . . . . . 9 (𝐴𝑉 → 1o ⊆ (wUniCl‘𝐴))
3125, 30unssd 4138 . . . . . . . 8 (𝐴𝑉 → (𝐴 ∪ 1o) ⊆ (wUniCl‘𝐴))
3224, 31eqsstrd 3981 . . . . . . 7 (𝐴𝑉 → (𝐹‘∅) ⊆ (wUniCl‘𝐴))
33 simplr 768 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → 𝑛 ∈ ω)
34 fvex 6656 . . . . . . . . . . . . 13 (𝐹𝑛) ∈ V
3534uniex 7442 . . . . . . . . . . . . 13 (𝐹𝑛) ∈ V
3634, 35unex 7444 . . . . . . . . . . . 12 ((𝐹𝑛) ∪ (𝐹𝑛)) ∈ V
37 prex 5306 . . . . . . . . . . . . . 14 {𝒫 𝑢, 𝑢} ∈ V
3834mptex 6959 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ∈ V
3938rnex 7592 . . . . . . . . . . . . . 14 ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ∈ V
4037, 39unex 7444 . . . . . . . . . . . . 13 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ∈ V
4134, 40iunex 7644 . . . . . . . . . . . 12 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ∈ V
4236, 41unex 7444 . . . . . . . . . . 11 (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))) ∈ V
43 id 22 . . . . . . . . . . . . . 14 (𝑤 = 𝑧𝑤 = 𝑧)
44 unieq 4822 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 𝑤 = 𝑧)
4543, 44uneq12d 4116 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑤 𝑤) = (𝑧 𝑧))
46 pweq 4528 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
47 unieq 4822 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 𝑢 = 𝑥)
4846, 47preq12d 4650 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → {𝒫 𝑢, 𝑢} = {𝒫 𝑥, 𝑥})
49 preq1 4642 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑥 → {𝑢, 𝑣} = {𝑥, 𝑣})
5049mpteq2dv 5135 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣𝑤 ↦ {𝑥, 𝑣}))
5150rneqd 5781 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣𝑤 ↦ {𝑥, 𝑣}))
5248, 51uneq12d 4116 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})))
5352cbviunv 4938 . . . . . . . . . . . . . 14 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣}))
54 preq2 4643 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑦 → {𝑥, 𝑣} = {𝑥, 𝑦})
5554cbvmptv 5142 . . . . . . . . . . . . . . . . . 18 (𝑣𝑤 ↦ {𝑥, 𝑣}) = (𝑦𝑤 ↦ {𝑥, 𝑦})
56 mpteq1 5127 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑦𝑤 ↦ {𝑥, 𝑦}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
5755, 56syl5eq 2868 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑣𝑤 ↦ {𝑥, 𝑣}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
5857rneqd 5781 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → ran (𝑣𝑤 ↦ {𝑥, 𝑣}) = ran (𝑦𝑧 ↦ {𝑥, 𝑦}))
5958uneq2d 4115 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6043, 59iuneq12d 4920 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6153, 60syl5eq 2868 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6245, 61uneq12d 4116 . . . . . . . . . . . 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 4822 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑛) → 𝑤 = (𝐹𝑛))
6563, 64uneq12d 4116 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑛) → (𝑤 𝑤) = ((𝐹𝑛) ∪ (𝐹𝑛)))
66 mpteq1 5127 . . . . . . . . . . . . . . . 16 (𝑤 = (𝐹𝑛) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))
6766rneqd 5781 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝑛) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))
6867uneq2d 4115 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑛) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})))
6963, 68iuneq12d 4920 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑛) → 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})))
7065, 69uneq12d 4116 . . . . . . . . . . . 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 8051 . . . . . . . . . . 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 589 . . . . . . . . . 10 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘suc 𝑛) = (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))))
73 simpr 488 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
7427ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (wUniCl‘𝐴) ∈ WUni)
7573sselda 3943 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
7674, 75wunelss 10107 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ⊆ (wUniCl‘𝐴))
7776ralrimiva 3170 . . . . . . . . . . . . 13 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹𝑛)𝑢 ⊆ (wUniCl‘𝐴))
78 unissb 4843 . . . . . . . . . . . . 13 ( (𝐹𝑛) ⊆ (wUniCl‘𝐴) ↔ ∀𝑢 ∈ (𝐹𝑛)𝑢 ⊆ (wUniCl‘𝐴))
7977, 78sylibr 237 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
8073, 79unssd 4138 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ((𝐹𝑛) ∪ (𝐹𝑛)) ⊆ (wUniCl‘𝐴))
8174, 75wunpw 10106 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝒫 𝑢 ∈ (wUniCl‘𝐴))
8274, 75wununi 10105 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
8381, 82prssd 4728 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → {𝒫 𝑢, 𝑢} ⊆ (wUniCl‘𝐴))
8474adantr 484 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → (wUniCl‘𝐴) ∈ WUni)
8575adantr 484 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
86 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
8786sselda 3943 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → 𝑣 ∈ (wUniCl‘𝐴))
8884, 85, 87wunpr 10108 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → {𝑢, 𝑣} ∈ (wUniCl‘𝐴))
8988fmpttd 6852 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}):(𝐹𝑛)⟶(wUniCl‘𝐴))
9089frnd 6494 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ⊆ (wUniCl‘𝐴))
9183, 90unssd 4138 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
9291ralrimiva 3170 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
93 iunss 4942 . . . . . . . . . . . 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 237 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
9580, 94unssd 4138 . . . . . . . . . 10 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))) ⊆ (wUniCl‘𝐴))
9672, 95eqsstrd 3981 . . . . . . . . 9 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))
9796ex 416 . . . . . . . 8 ((𝐴𝑉𝑛 ∈ ω) → ((𝐹𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
9897expcom 417 . . . . . . 7 (𝑛 ∈ ω → (𝐴𝑉 → ((𝐹𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))))
9913, 15, 17, 32, 98finds2 7585 . . . . . 6 (𝑚 ∈ ω → (𝐴𝑉 → (𝐹𝑚) ⊆ (wUniCl‘𝐴)))
10099com12 32 . . . . 5 (𝐴𝑉 → (𝑚 ∈ ω → (𝐹𝑚) ⊆ (wUniCl‘𝐴)))
101100ralrimiv 3169 . . . 4 (𝐴𝑉 → ∀𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
102 iunss 4942 . . . 4 ( 𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ ∀𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
103101, 102sylibr 237 . . 3 (𝐴𝑉 𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
10411, 103eqsstrid 3991 . 2 (𝐴𝑉𝑈 ⊆ (wUniCl‘𝐴))
1055, 104eqssd 3960 1 (𝐴𝑉 → (wUniCl‘𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3126  Vcvv 3471  cun 3908  wss 3910  c0 4266  𝒫 cpw 4512  {csn 4540  {cpr 4542   cuni 4811   ciun 4892  cmpt 5119  ran crn 5529  cres 5530  Oncon0 6164  suc csuc 6166   Fn wfn 6323  cfv 6328  ωcom 7555  reccrdg 8020  1oc1o 8070  WUnicwun 10099  wUniClcwunm 10100
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-wun 10101  df-wunc 10102
This theorem is referenced by: (None)
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