wuncval2 - Metamath Proof Explorer

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Theorem wuncval2 10670

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) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1_o)) ↾ ω)
wuncval2.u	⊢ 𝑈 = ∪ ran 𝐹

**Assertion**
Ref	Expression
wuncval2	⊢ (𝐴 ∈ 𝑉 → (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.

Step	Hyp	Ref	Expression
1		wuncval2.f	. . . 4 ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1_o)) ↾ ω)
2		wuncval2.u	. . . 4 ⊢ 𝑈 = ∪ ran 𝐹
3	1, 2	wunex2 10661	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))
4		wuncss 10668	. . 3 ⊢ ((𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈) → (wUniCl‘𝐴) ⊆ 𝑈)
5	3, 4	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ⊆ 𝑈)
6		frfnom 8376	. . . . . 6 ⊢ (rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1_o)) ↾ ω) Fn ω
7	1	fneq1i 6597	. . . . . 6 ⊢ (𝐹 Fn ω ↔ (rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1_o)) ↾ ω) Fn ω)
8	6, 7	mpbir 231	. . . . 5 ⊢ 𝐹 Fn ω
9		fniunfv 7203	. . . . 5 ⊢ (𝐹 Fn ω → ∪ 𝑚 ∈ ω (𝐹‘𝑚) = ∪ ran 𝐹)
10	8, 9	ax-mp 5	. . . 4 ⊢ ∪ 𝑚 ∈ ω (𝐹‘𝑚) = ∪ ran 𝐹
11	2, 10	eqtr4i 2763	. . 3 ⊢ 𝑈 = ∪ 𝑚 ∈ ω (𝐹‘𝑚)
12		fveq2 6842	. . . . . . . 8 ⊢ (𝑚 = ∅ → (𝐹‘𝑚) = (𝐹‘∅))
13	12	sseq1d 3967	. . . . . . 7 ⊢ (𝑚 = ∅ → ((𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘∅) ⊆ (wUniCl‘𝐴)))
14		fveq2 6842	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
15	14	sseq1d 3967	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)))
16		fveq2 6842	. . . . . . . 8 ⊢ (𝑚 = suc 𝑛 → (𝐹‘𝑚) = (𝐹‘suc 𝑛))
17	16	sseq1d 3967	. . . . . . 7 ⊢ (𝑚 = suc 𝑛 → ((𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
18		1on 8419	. . . . . . . . . 10 ⊢ 1_o ∈ On
19		unexg 7698	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 1_o ∈ On) → (𝐴 ∪ 1_o) ∈ V)
20	18, 19	mpan2 692	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 1_o) ∈ V)
21	1	fveq1i 6843	. . . . . . . . . 10 ⊢ (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1_o)) ↾ ω)‘∅)
22		fr0g 8377	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 1_o) ∈ V → ((rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1_o)) ↾ ω)‘∅) = (𝐴 ∪ 1_o))
23	21, 22	eqtrid 2784	. . . . . . . . 9 ⊢ ((𝐴 ∪ 1_o) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1_o))
24	20, 23	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘∅) = (𝐴 ∪ 1_o))
25		wuncid 10666	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (wUniCl‘𝐴))
26		df1o2 8414	. . . . . . . . . 10 ⊢ 1_o = {∅}
27		wunccl 10667	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ∈ WUni)
28	27	wun0 10641	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ (wUniCl‘𝐴))
29	28	snssd 4767	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → {∅} ⊆ (wUniCl‘𝐴))
30	26, 29	eqsstrid 3974	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 1_o ⊆ (wUniCl‘𝐴))
31	25, 30	unssd 4146	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 1_o) ⊆ (wUniCl‘𝐴))
32	24, 31	eqsstrd 3970	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘∅) ⊆ (wUniCl‘𝐴))
33		simplr 769	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → 𝑛 ∈ ω)
34		fvex 6855	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑛) ∈ V
35	34	uniex 7696	. . . . . . . . . . . . 13 ⊢ ∪ (𝐹‘𝑛) ∈ V
36	34, 35	unex 7699	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ∈ V
37		prex 5384	. . . . . . . . . . . . . 14 ⊢ {𝒫 𝑢, ∪ 𝑢} ∈ V
38	34	mptex 7179	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}) ∈ V
39	38	rnex 7862	. . . . . . . . . . . . . 14 ⊢ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}) ∈ V
40	37, 39	unex 7699	. . . . . . . . . . . . 13 ⊢ ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ∈ V
41	34, 40	iunex 7922	. . . . . . . . . . . 12 ⊢ ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ∈ V
42	36, 41	unex 7699	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ∪ ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))) ∈ V
43		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → 𝑤 = 𝑧)
44		unieq 4876	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → ∪ 𝑤 = ∪ 𝑧)
45	43, 44	uneq12d 4123	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑤 ∪ ∪ 𝑤) = (𝑧 ∪ ∪ 𝑧))
46		pweq 4570	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
47		unieq 4876	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑥 → ∪ 𝑢 = ∪ 𝑥)
48	46, 47	preq12d 4700	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → {𝒫 𝑢, ∪ 𝑢} = {𝒫 𝑥, ∪ 𝑥})
49		preq1 4692	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑥 → {𝑢, 𝑣} = {𝑥, 𝑣})
50	49	mpteq2dv 5194	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑥 → (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}))
51	50	rneqd 5895	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}))
52	48, 51	uneq12d 4123	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣})))
53	52	cbviunv 4996	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ∪ 𝑥 ∈ 𝑤 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}))
54		preq2 4693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑦 → {𝑥, 𝑣} = {𝑥, 𝑦})
55	54	cbvmptv 5204	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}) = (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦})
56		mpteq1 5189	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦}) = (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))
57	55, 56	eqtrid 2784	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}) = (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))
58	57	rneqd 5895	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}) = ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))
59	58	uneq2d 4122	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣})) = ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))
60	43, 59	iuneq12d 4978	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → ∪ 𝑥 ∈ 𝑤 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣})) = ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))
61	53, 60	eqtrid 2784	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))
62	45, 61	uneq12d 4123	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → ((𝑤 ∪ ∪ 𝑤) ∪ ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}))) = ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))))
63		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐹‘𝑛) → 𝑤 = (𝐹‘𝑛))
64		unieq 4876	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐹‘𝑛) → ∪ 𝑤 = ∪ (𝐹‘𝑛))
65	63, 64	uneq12d 4123	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐹‘𝑛) → (𝑤 ∪ ∪ 𝑤) = ((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)))
66		mpteq1 5189	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝐹‘𝑛) → (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))
67	66	rneqd 5895	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐹‘𝑛) → ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))
68	67	uneq2d 4122	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐹‘𝑛) → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})))
69	63, 68	iuneq12d 4978	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐹‘𝑛) → ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})))
70	65, 69	uneq12d 4123	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹‘𝑛) → ((𝑤 ∪ ∪ 𝑤) ∪ ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}))) = (((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ∪ ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))))
71	1, 62, 70	frsucmpt2 8381	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ω ∧ (((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ∪ ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))) ∈ V) → (𝐹‘suc 𝑛) = (((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ∪ ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))))
72	33, 42, 71	sylancl 587	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘suc 𝑛) = (((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ∪ ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))))
73		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘𝑛) ⊆ (wUniCl‘𝐴))
74	27	ad3antrrr 731	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → (wUniCl‘𝐴) ∈ WUni)
75	73	sselda 3935	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
76	74, 75	wunelss 10631	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → 𝑢 ⊆ (wUniCl‘𝐴))
77	76	ralrimiva 3130	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹‘𝑛)𝑢 ⊆ (wUniCl‘𝐴))
78		unissb 4898	. . . . . . . . . . . . 13 ⊢ (∪ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴) ↔ ∀𝑢 ∈ (𝐹‘𝑛)𝑢 ⊆ (wUniCl‘𝐴))
79	77, 78	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ∪ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴))
80	73, 79	unssd 4146	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ⊆ (wUniCl‘𝐴))
81	74, 75	wunpw 10630	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → 𝒫 𝑢 ∈ (wUniCl‘𝐴))
82	74, 75	wununi 10629	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → ∪ 𝑢 ∈ (wUniCl‘𝐴))
83	81, 82	prssd 4780	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → {𝒫 𝑢, ∪ 𝑢} ⊆ (wUniCl‘𝐴))
84	74	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → (wUniCl‘𝐴) ∈ WUni)
85	75	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
86		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → (𝐹‘𝑛) ⊆ (wUniCl‘𝐴))
87	86	sselda 3935	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → 𝑣 ∈ (wUniCl‘𝐴))
88	84, 85, 87	wunpr 10632	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → {𝑢, 𝑣} ∈ (wUniCl‘𝐴))
89	88	fmpttd 7069	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}):(𝐹‘𝑛)⟶(wUniCl‘𝐴))
90	89	frnd 6678	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}) ⊆ (wUniCl‘𝐴))
91	83, 90	unssd 4146	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
92	91	ralrimiva 3130	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
93		iunss 5002	. . . . . . . . . . . 12 ⊢ (∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴) ↔ ∀𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
94	92, 93	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
95	80, 94	unssd 4146	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → (((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ∪ ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))) ⊆ (wUniCl‘𝐴))
96	72, 95	eqsstrd 3970	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))
97	96	ex 412	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
98	97	expcom 413	. . . . . . 7 ⊢ (𝑛 ∈ ω → (𝐴 ∈ 𝑉 → ((𝐹‘𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))))
99	13, 15, 17, 32, 98	finds2 7850	. . . . . 6 ⊢ (𝑚 ∈ ω → (𝐴 ∈ 𝑉 → (𝐹‘𝑚) ⊆ (wUniCl‘𝐴)))
100	99	com12 32	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑚 ∈ ω → (𝐹‘𝑚) ⊆ (wUniCl‘𝐴)))
101	100	ralrimiv 3129	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴))
102		iunss 5002	. . . 4 ⊢ (∪ 𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ ∀𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴))
103	101, 102	sylibr 234	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴))
104	11, 103	eqsstrid 3974	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑈 ⊆ (wUniCl‘𝐴))
105	5, 104	eqssd 3953	1 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3442 ∪ cun 3901 ⊆ wss 3903 ∅c0 4287 𝒫 cpw 4556 {csn 4582 {cpr 4584 ∪ cuni 4865 ∪ ciun 4948 ↦ cmpt 5181 ran crn 5633 ↾ cres 5634 Oncon0 6325 suc csuc 6327 Fn wfn 6495 ‘cfv 6500 ωcom 7818 reccrdg 8350 1_oc1o 8400 WUnicwun 10623 wUniClcwunm 10624

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 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-wun 10625 df-wunc 10626

This theorem is referenced by: (None)

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