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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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