MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wuncval2 Structured version   Visualization version   GIF version

Theorem wuncval2 10748
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 10739 . . 3 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
4 wuncss 10746 . . 3 ((𝑈 ∈ WUni ∧ 𝐴𝑈) → (wUniCl‘𝐴) ⊆ 𝑈)
53, 4syl 17 . 2 (𝐴𝑉 → (wUniCl‘𝐴) ⊆ 𝑈)
6 frfnom 8441 . . . . . 6 (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) Fn ω
71fneq1i 6646 . . . . . 6 (𝐹 Fn ω ↔ (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) Fn ω)
86, 7mpbir 230 . . . . 5 𝐹 Fn ω
9 fniunfv 7249 . . . . 5 (𝐹 Fn ω → 𝑚 ∈ ω (𝐹𝑚) = ran 𝐹)
108, 9ax-mp 5 . . . 4 𝑚 ∈ ω (𝐹𝑚) = ran 𝐹
112, 10eqtr4i 2762 . . 3 𝑈 = 𝑚 ∈ ω (𝐹𝑚)
12 fveq2 6891 . . . . . . . 8 (𝑚 = ∅ → (𝐹𝑚) = (𝐹‘∅))
1312sseq1d 4013 . . . . . . 7 (𝑚 = ∅ → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘∅) ⊆ (wUniCl‘𝐴)))
14 fveq2 6891 . . . . . . . 8 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
1514sseq1d 4013 . . . . . . 7 (𝑚 = 𝑛 → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹𝑛) ⊆ (wUniCl‘𝐴)))
16 fveq2 6891 . . . . . . . 8 (𝑚 = suc 𝑛 → (𝐹𝑚) = (𝐹‘suc 𝑛))
1716sseq1d 4013 . . . . . . 7 (𝑚 = suc 𝑛 → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
18 1on 8484 . . . . . . . . . 10 1o ∈ On
19 unexg 7740 . . . . . . . . . 10 ((𝐴𝑉 ∧ 1o ∈ On) → (𝐴 ∪ 1o) ∈ V)
2018, 19mpan2 688 . . . . . . . . 9 (𝐴𝑉 → (𝐴 ∪ 1o) ∈ V)
211fveq1i 6892 . . . . . . . . . 10 (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅)
22 fr0g 8442 . . . . . . . . . 10 ((𝐴 ∪ 1o) ∈ V → ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)‘∅) = (𝐴 ∪ 1o))
2321, 22eqtrid 2783 . . . . . . . . 9 ((𝐴 ∪ 1o) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1o))
2420, 23syl 17 . . . . . . . 8 (𝐴𝑉 → (𝐹‘∅) = (𝐴 ∪ 1o))
25 wuncid 10744 . . . . . . . . 9 (𝐴𝑉𝐴 ⊆ (wUniCl‘𝐴))
26 df1o2 8479 . . . . . . . . . 10 1o = {∅}
27 wunccl 10745 . . . . . . . . . . . 12 (𝐴𝑉 → (wUniCl‘𝐴) ∈ WUni)
2827wun0 10719 . . . . . . . . . . 11 (𝐴𝑉 → ∅ ∈ (wUniCl‘𝐴))
2928snssd 4812 . . . . . . . . . 10 (𝐴𝑉 → {∅} ⊆ (wUniCl‘𝐴))
3026, 29eqsstrid 4030 . . . . . . . . 9 (𝐴𝑉 → 1o ⊆ (wUniCl‘𝐴))
3125, 30unssd 4186 . . . . . . . 8 (𝐴𝑉 → (𝐴 ∪ 1o) ⊆ (wUniCl‘𝐴))
3224, 31eqsstrd 4020 . . . . . . 7 (𝐴𝑉 → (𝐹‘∅) ⊆ (wUniCl‘𝐴))
33 simplr 766 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → 𝑛 ∈ ω)
34 fvex 6904 . . . . . . . . . . . . 13 (𝐹𝑛) ∈ V
3534uniex 7735 . . . . . . . . . . . . 13 (𝐹𝑛) ∈ V
3634, 35unex 7737 . . . . . . . . . . . 12 ((𝐹𝑛) ∪ (𝐹𝑛)) ∈ V
37 prex 5432 . . . . . . . . . . . . . 14 {𝒫 𝑢, 𝑢} ∈ V
3834mptex 7227 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ∈ V
3938rnex 7907 . . . . . . . . . . . . . 14 ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ∈ V
4037, 39unex 7737 . . . . . . . . . . . . 13 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ∈ V
4134, 40iunex 7959 . . . . . . . . . . . 12 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ∈ V
4236, 41unex 7737 . . . . . . . . . . 11 (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))) ∈ V
43 id 22 . . . . . . . . . . . . . 14 (𝑤 = 𝑧𝑤 = 𝑧)
44 unieq 4919 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 𝑤 = 𝑧)
4543, 44uneq12d 4164 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑤 𝑤) = (𝑧 𝑧))
46 pweq 4616 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
47 unieq 4919 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 𝑢 = 𝑥)
4846, 47preq12d 4745 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → {𝒫 𝑢, 𝑢} = {𝒫 𝑥, 𝑥})
49 preq1 4737 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑥 → {𝑢, 𝑣} = {𝑥, 𝑣})
5049mpteq2dv 5250 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣𝑤 ↦ {𝑥, 𝑣}))
5150rneqd 5937 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣𝑤 ↦ {𝑥, 𝑣}))
5248, 51uneq12d 4164 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})))
5352cbviunv 5043 . . . . . . . . . . . . . 14 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣}))
54 preq2 4738 . . . . . . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑣𝑤 ↦ {𝑥, 𝑣}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
5857rneqd 5937 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → ran (𝑣𝑤 ↦ {𝑥, 𝑣}) = ran (𝑦𝑧 ↦ {𝑥, 𝑦}))
5958uneq2d 4163 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6043, 59iuneq12d 5025 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6153, 60eqtrid 2783 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6245, 61uneq12d 4164 . . . . . . . . . . . 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 4919 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑛) → 𝑤 = (𝐹𝑛))
6563, 64uneq12d 4164 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑛) → (𝑤 𝑤) = ((𝐹𝑛) ∪ (𝐹𝑛)))
66 mpteq1 5241 . . . . . . . . . . . . . . . 16 (𝑤 = (𝐹𝑛) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))
6766rneqd 5937 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝑛) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))
6867uneq2d 4163 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑛) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})))
6963, 68iuneq12d 5025 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑛) → 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})))
7065, 69uneq12d 4164 . . . . . . . . . . . 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 8446 . . . . . . . . . . 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 585 . . . . . . . . . 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 727 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (wUniCl‘𝐴) ∈ WUni)
7573sselda 3982 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
7674, 75wunelss 10709 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ⊆ (wUniCl‘𝐴))
7776ralrimiva 3145 . . . . . . . . . . . . 13 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹𝑛)𝑢 ⊆ (wUniCl‘𝐴))
78 unissb 4943 . . . . . . . . . . . . 13 ( (𝐹𝑛) ⊆ (wUniCl‘𝐴) ↔ ∀𝑢 ∈ (𝐹𝑛)𝑢 ⊆ (wUniCl‘𝐴))
7977, 78sylibr 233 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
8073, 79unssd 4186 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ((𝐹𝑛) ∪ (𝐹𝑛)) ⊆ (wUniCl‘𝐴))
8174, 75wunpw 10708 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝒫 𝑢 ∈ (wUniCl‘𝐴))
8274, 75wununi 10707 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
8381, 82prssd 4825 . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
8786sselda 3982 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → 𝑣 ∈ (wUniCl‘𝐴))
8884, 85, 87wunpr 10710 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → {𝑢, 𝑣} ∈ (wUniCl‘𝐴))
8988fmpttd 7116 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}):(𝐹𝑛)⟶(wUniCl‘𝐴))
9089frnd 6725 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ⊆ (wUniCl‘𝐴))
9183, 90unssd 4186 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
9291ralrimiva 3145 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
93 iunss 5048 . . . . . . . . . . . 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 233 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
9580, 94unssd 4186 . . . . . . . . . 10 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))) ⊆ (wUniCl‘𝐴))
9672, 95eqsstrd 4020 . . . . . . . . 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 7895 . . . . . 6 (𝑚 ∈ ω → (𝐴𝑉 → (𝐹𝑚) ⊆ (wUniCl‘𝐴)))
10099com12 32 . . . . 5 (𝐴𝑉 → (𝑚 ∈ ω → (𝐹𝑚) ⊆ (wUniCl‘𝐴)))
101100ralrimiv 3144 . . . 4 (𝐴𝑉 → ∀𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
102 iunss 5048 . . . 4 ( 𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ ∀𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
103101, 102sylibr 233 . . 3 (𝐴𝑉 𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
10411, 103eqsstrid 4030 . 2 (𝐴𝑉𝑈 ⊆ (wUniCl‘𝐴))
1055, 104eqssd 3999 1 (𝐴𝑉 → (wUniCl‘𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wral 3060  Vcvv 3473  cun 3946  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628  {cpr 4630   cuni 4908   ciun 4997  cmpt 5231  ran crn 5677  cres 5678  Oncon0 6364  suc csuc 6366   Fn wfn 6538  cfv 6543  ωcom 7859  reccrdg 8415  1oc1o 8465  WUnicwun 10701  wUniClcwunm 10702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-wun 10703  df-wunc 10704
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator