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Theorem wuncval2 9607
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 9598 . . 3 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
4 wuncss 9605 . . 3 ((𝑈 ∈ WUni ∧ 𝐴𝑈) → (wUniCl‘𝐴) ⊆ 𝑈)
53, 4syl 17 . 2 (𝐴𝑉 → (wUniCl‘𝐴) ⊆ 𝑈)
6 frfnom 7575 . . . . . 6 (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω) Fn ω
71fneq1i 6023 . . . . . 6 (𝐹 Fn ω ↔ (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω) Fn ω)
86, 7mpbir 221 . . . . 5 𝐹 Fn ω
9 fniunfv 6545 . . . . 5 (𝐹 Fn ω → 𝑚 ∈ ω (𝐹𝑚) = ran 𝐹)
108, 9ax-mp 5 . . . 4 𝑚 ∈ ω (𝐹𝑚) = ran 𝐹
112, 10eqtr4i 2676 . . 3 𝑈 = 𝑚 ∈ ω (𝐹𝑚)
12 fveq2 6229 . . . . . . . 8 (𝑚 = ∅ → (𝐹𝑚) = (𝐹‘∅))
1312sseq1d 3665 . . . . . . 7 (𝑚 = ∅ → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘∅) ⊆ (wUniCl‘𝐴)))
14 fveq2 6229 . . . . . . . 8 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
1514sseq1d 3665 . . . . . . 7 (𝑚 = 𝑛 → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹𝑛) ⊆ (wUniCl‘𝐴)))
16 fveq2 6229 . . . . . . . 8 (𝑚 = suc 𝑛 → (𝐹𝑚) = (𝐹‘suc 𝑛))
1716sseq1d 3665 . . . . . . 7 (𝑚 = suc 𝑛 → ((𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
18 1on 7612 . . . . . . . . . 10 1𝑜 ∈ On
19 unexg 7001 . . . . . . . . . 10 ((𝐴𝑉 ∧ 1𝑜 ∈ On) → (𝐴 ∪ 1𝑜) ∈ V)
2018, 19mpan2 707 . . . . . . . . 9 (𝐴𝑉 → (𝐴 ∪ 1𝑜) ∈ V)
211fveq1i 6230 . . . . . . . . . 10 (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω)‘∅)
22 fr0g 7576 . . . . . . . . . 10 ((𝐴 ∪ 1𝑜) ∈ V → ((rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω)‘∅) = (𝐴 ∪ 1𝑜))
2321, 22syl5eq 2697 . . . . . . . . 9 ((𝐴 ∪ 1𝑜) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1𝑜))
2420, 23syl 17 . . . . . . . 8 (𝐴𝑉 → (𝐹‘∅) = (𝐴 ∪ 1𝑜))
25 wuncid 9603 . . . . . . . . 9 (𝐴𝑉𝐴 ⊆ (wUniCl‘𝐴))
26 df1o2 7617 . . . . . . . . . 10 1𝑜 = {∅}
27 wunccl 9604 . . . . . . . . . . . 12 (𝐴𝑉 → (wUniCl‘𝐴) ∈ WUni)
2827wun0 9578 . . . . . . . . . . 11 (𝐴𝑉 → ∅ ∈ (wUniCl‘𝐴))
2928snssd 4372 . . . . . . . . . 10 (𝐴𝑉 → {∅} ⊆ (wUniCl‘𝐴))
3026, 29syl5eqss 3682 . . . . . . . . 9 (𝐴𝑉 → 1𝑜 ⊆ (wUniCl‘𝐴))
3125, 30unssd 3822 . . . . . . . 8 (𝐴𝑉 → (𝐴 ∪ 1𝑜) ⊆ (wUniCl‘𝐴))
3224, 31eqsstrd 3672 . . . . . . 7 (𝐴𝑉 → (𝐹‘∅) ⊆ (wUniCl‘𝐴))
33 simplr 807 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → 𝑛 ∈ ω)
34 fvex 6239 . . . . . . . . . . . . 13 (𝐹𝑛) ∈ V
3534uniex 6995 . . . . . . . . . . . . 13 (𝐹𝑛) ∈ V
3634, 35unex 6998 . . . . . . . . . . . 12 ((𝐹𝑛) ∪ (𝐹𝑛)) ∈ V
37 prex 4939 . . . . . . . . . . . . . 14 {𝒫 𝑢, 𝑢} ∈ V
3834mptex 6527 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ∈ V
3938rnex 7142 . . . . . . . . . . . . . 14 ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ∈ V
4037, 39unex 6998 . . . . . . . . . . . . 13 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ∈ V
4134, 40iunex 7189 . . . . . . . . . . . 12 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ∈ V
4236, 41unex 6998 . . . . . . . . . . 11 (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))) ∈ V
43 id 22 . . . . . . . . . . . . . 14 (𝑤 = 𝑧𝑤 = 𝑧)
44 unieq 4476 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 𝑤 = 𝑧)
4543, 44uneq12d 3801 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑤 𝑤) = (𝑧 𝑧))
46 pweq 4194 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
47 unieq 4476 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 𝑢 = 𝑥)
4846, 47preq12d 4308 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → {𝒫 𝑢, 𝑢} = {𝒫 𝑥, 𝑥})
49 preq1 4300 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑥 → {𝑢, 𝑣} = {𝑥, 𝑣})
5049mpteq2dv 4778 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑥 → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣𝑤 ↦ {𝑥, 𝑣}))
5150rneqd 5385 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣𝑤 ↦ {𝑥, 𝑣}))
5248, 51uneq12d 3801 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})))
5352cbviunv 4591 . . . . . . . . . . . . . 14 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣}))
54 preq2 4301 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑦 → {𝑥, 𝑣} = {𝑥, 𝑦})
5554cbvmptv 4783 . . . . . . . . . . . . . . . . . 18 (𝑣𝑤 ↦ {𝑥, 𝑣}) = (𝑦𝑤 ↦ {𝑥, 𝑦})
56 mpteq1 4770 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑦𝑤 ↦ {𝑥, 𝑦}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
5755, 56syl5eq 2697 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑣𝑤 ↦ {𝑥, 𝑣}) = (𝑦𝑧 ↦ {𝑥, 𝑦}))
5857rneqd 5385 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → ran (𝑣𝑤 ↦ {𝑥, 𝑣}) = ran (𝑦𝑧 ↦ {𝑥, 𝑦}))
5958uneq2d 3800 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})) = ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6043, 59iuneq12d 4578 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 𝑥𝑤 ({𝒫 𝑥, 𝑥} ∪ ran (𝑣𝑤 ↦ {𝑥, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6153, 60syl5eq 2697 . . . . . . . . . . . . 13 (𝑤 = 𝑧 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))
6245, 61uneq12d 3801 . . . . . . . . . . . 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 4476 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑛) → 𝑤 = (𝐹𝑛))
6563, 64uneq12d 3801 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑛) → (𝑤 𝑤) = ((𝐹𝑛) ∪ (𝐹𝑛)))
66 mpteq1 4770 . . . . . . . . . . . . . . . 16 (𝑤 = (𝐹𝑛) → (𝑣𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))
6766rneqd 5385 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝑛) → ran (𝑣𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))
6867uneq2d 3800 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝑛) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})))
6963, 68iuneq12d 4578 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑛) → 𝑢𝑤 ({𝒫 𝑢, 𝑢} ∪ ran (𝑣𝑤 ↦ {𝑢, 𝑣})) = 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})))
7065, 69uneq12d 3801 . . . . . . . . . . . 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 7580 . . . . . . . . . . 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 695 . . . . . . . . . 10 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘suc 𝑛) = (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))))
73 simpr 476 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
7427ad3antrrr 766 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (wUniCl‘𝐴) ∈ WUni)
7573sselda 3636 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
7674, 75wunelss 9568 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ⊆ (wUniCl‘𝐴))
7776ralrimiva 2995 . . . . . . . . . . . . 13 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹𝑛)𝑢 ⊆ (wUniCl‘𝐴))
78 unissb 4501 . . . . . . . . . . . . 13 ( (𝐹𝑛) ⊆ (wUniCl‘𝐴) ↔ ∀𝑢 ∈ (𝐹𝑛)𝑢 ⊆ (wUniCl‘𝐴))
7977, 78sylibr 224 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
8073, 79unssd 3822 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ((𝐹𝑛) ∪ (𝐹𝑛)) ⊆ (wUniCl‘𝐴))
8174, 75wunpw 9567 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝒫 𝑢 ∈ (wUniCl‘𝐴))
8274, 75wununi 9566 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
83 prssi 4385 . . . . . . . . . . . . . . 15 ((𝒫 𝑢 ∈ (wUniCl‘𝐴) ∧ 𝑢 ∈ (wUniCl‘𝐴)) → {𝒫 𝑢, 𝑢} ⊆ (wUniCl‘𝐴))
8481, 82, 83syl2anc 694 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → {𝒫 𝑢, 𝑢} ⊆ (wUniCl‘𝐴))
8574adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → (wUniCl‘𝐴) ∈ WUni)
8675adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
87 simplr 807 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (𝐹𝑛) ⊆ (wUniCl‘𝐴))
8887sselda 3636 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → 𝑣 ∈ (wUniCl‘𝐴))
8985, 86, 88wunpr 9569 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) ∧ 𝑣 ∈ (𝐹𝑛)) → {𝑢, 𝑣} ∈ (wUniCl‘𝐴))
90 eqid 2651 . . . . . . . . . . . . . . . 16 (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})
9189, 90fmptd 6425 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}):(𝐹𝑛)⟶(wUniCl‘𝐴))
92 frn 6091 . . . . . . . . . . . . . . 15 ((𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}):(𝐹𝑛)⟶(wUniCl‘𝐴) → ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ⊆ (wUniCl‘𝐴))
9391, 92syl 17 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}) ⊆ (wUniCl‘𝐴))
9484, 93unssd 3822 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹𝑛)) → ({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
9594ralrimiva 2995 . . . . . . . . . . . 12 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
96 iunss 4593 . . . . . . . . . . . 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))
9795, 96sylibr 224 . . . . . . . . . . 11 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
9880, 97unssd 3822 . . . . . . . . . 10 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (((𝐹𝑛) ∪ (𝐹𝑛)) ∪ 𝑢 ∈ (𝐹𝑛)({𝒫 𝑢, 𝑢} ∪ ran (𝑣 ∈ (𝐹𝑛) ↦ {𝑢, 𝑣}))) ⊆ (wUniCl‘𝐴))
9972, 98eqsstrd 3672 . . . . . . . . 9 (((𝐴𝑉𝑛 ∈ ω) ∧ (𝐹𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))
10099ex 449 . . . . . . . 8 ((𝐴𝑉𝑛 ∈ ω) → ((𝐹𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
101100expcom 450 . . . . . . 7 (𝑛 ∈ ω → (𝐴𝑉 → ((𝐹𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))))
10213, 15, 17, 32, 101finds2 7136 . . . . . 6 (𝑚 ∈ ω → (𝐴𝑉 → (𝐹𝑚) ⊆ (wUniCl‘𝐴)))
103102com12 32 . . . . 5 (𝐴𝑉 → (𝑚 ∈ ω → (𝐹𝑚) ⊆ (wUniCl‘𝐴)))
104103ralrimiv 2994 . . . 4 (𝐴𝑉 → ∀𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
105 iunss 4593 . . . 4 ( 𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴) ↔ ∀𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
106104, 105sylibr 224 . . 3 (𝐴𝑉 𝑚 ∈ ω (𝐹𝑚) ⊆ (wUniCl‘𝐴))
10711, 106syl5eqss 3682 . 2 (𝐴𝑉𝑈 ⊆ (wUniCl‘𝐴))
1085, 107eqssd 3653 1 (𝐴𝑉 → (wUniCl‘𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cun 3605  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  {cpr 4212   cuni 4468   ciun 4552  cmpt 4762  ran crn 5144  cres 5145  Oncon0 5761  suc csuc 5763   Fn wfn 5921  wf 5922  cfv 5926  ωcom 7107  reccrdg 7550  1𝑜c1o 7598  WUnicwun 9560  wUniClcwunm 9561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-wun 9562  df-wunc 9563
This theorem is referenced by: (None)
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