wuncval2 - Metamath Proof Explorer

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Theorem wuncval2 10733

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 10724	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))
4		wuncss 10731	. . 3 ⊢ ((𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈) → (wUniCl‘𝐴) ⊆ 𝑈)
5	3, 4	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ⊆ 𝑈)
6		frfnom 8423	. . . . . 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 6634	. . . . . 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 234	. . . . 5 ⊢ 𝐹 Fn ω
9		fniunfv 7247	. . . . 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 2789	. . 3 ⊢ 𝑈 = ∪ 𝑚 ∈ ω (𝐹‘𝑚)
12		fveq2 6883	. . . . . . . 8 ⊢ (𝑚 = ∅ → (𝐹‘𝑚) = (𝐹‘∅))
13	12	sseq1d 3969	. . . . . . 7 ⊢ (𝑚 = ∅ → ((𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘∅) ⊆ (wUniCl‘𝐴)))
14		fveq2 6883	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
15	14	sseq1d 3969	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)))
16		fveq2 6883	. . . . . . . 8 ⊢ (𝑚 = suc 𝑛 → (𝐹‘𝑚) = (𝐹‘suc 𝑛))
17	16	sseq1d 3969	. . . . . . 7 ⊢ (𝑚 = suc 𝑛 → ((𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
18		1on 8467	. . . . . . . . . 10 ⊢ 1_o ∈ On
19		unexg 7743	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 1_o ∈ On) → (𝐴 ∪ 1_o) ∈ V)
20	18, 19	mpan2 703	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 1_o) ∈ V)
21	1	fveq1i 6884	. . . . . . . . . 10 ⊢ (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1_o)) ↾ ω)‘∅)
22		fr0g 8424	. . . . . . . . . 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 2810	. . . . . . . . 9 ⊢ ((𝐴 ∪ 1_o) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1_o))
24	20, 23	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘∅) = (𝐴 ∪ 1_o))
25		wuncid 10729	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (wUniCl‘𝐴))
26		df1o2 8461	. . . . . . . . . 10 ⊢ 1_o = {∅}
27		wunccl 10730	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ∈ WUni)
28	27	wun0 10704	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ (wUniCl‘𝐴))
29	28	snssd 4753	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → {∅} ⊆ (wUniCl‘𝐴))
30	26, 29	eqsstrid 3976	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 1_o ⊆ (wUniCl‘𝐴))
31	25, 30	unssd 4146	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 1_o) ⊆ (wUniCl‘𝐴))
32	24, 31	eqsstrd 3972	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘∅) ⊆ (wUniCl‘𝐴))
33		simplr 780	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → 𝑛 ∈ ω)
34		fvex 6896	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑛) ∈ V
35	34	uniex 7741	. . . . . . . . . . . . 13 ⊢ ∪ (𝐹‘𝑛) ∈ V
36	34, 35	unex 7744	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ∈ V
37		prex 5411	. . . . . . . . . . . . . 14 ⊢ {𝒫 𝑢, ∪ 𝑢} ∈ V
38	34	mptex 7223	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}) ∈ V
39	38	rnex 7908	. . . . . . . . . . . . . 14 ⊢ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}) ∈ V
40	37, 39	unex 7744	. . . . . . . . . . . . 13 ⊢ ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ∈ V
41	34, 40	iunex 7966	. . . . . . . . . . . 12 ⊢ ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ∈ V
42	36, 41	unex 7744	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ∪ ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))) ∈ V
43		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → 𝑤 = 𝑧)
44		unieq 4884	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → ∪ 𝑤 = ∪ 𝑧)
45	43, 44	uneq12d 4124	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑤 ∪ ∪ 𝑤) = (𝑧 ∪ ∪ 𝑧))
46		pweq 4577	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
47		unieq 4884	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑥 → ∪ 𝑢 = ∪ 𝑥)
48	46, 47	preq12d 4708	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → {𝒫 𝑢, ∪ 𝑢} = {𝒫 𝑥, ∪ 𝑥})
49		preq1 4700	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑥 → {𝑢, 𝑣} = {𝑥, 𝑣})
50	49	mpteq2dv 5206	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑥 → (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}))
51	50	rneqd 5930	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}))
52	48, 51	uneq12d 4124	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣})))
53	52	cbviunv 5004	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ∪ 𝑥 ∈ 𝑤 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}))
54		preq2 4701	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑦 → {𝑥, 𝑣} = {𝑥, 𝑦})
55	54	cbvmptv 5216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}) = (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦})
56		mpteq1 5201	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦}) = (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))
57	55, 56	eqtrid 2810	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}) = (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))
58	57	rneqd 5930	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}) = ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))
59	58	uneq2d 4123	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣})) = ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))
60	43, 59	iuneq12d 4987	. . . . . . . . . . . . . 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 2810	. . . . . . . . . . . . 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 4124	. . . . . . . . . . . 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 23	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐹‘𝑛) → 𝑤 = (𝐹‘𝑛))
64		unieq 4884	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐹‘𝑛) → ∪ 𝑤 = ∪ (𝐹‘𝑛))
65	63, 64	uneq12d 4124	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐹‘𝑛) → (𝑤 ∪ ∪ 𝑤) = ((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)))
66		mpteq1 5201	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝐹‘𝑛) → (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))
67	66	rneqd 5930	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐹‘𝑛) → ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))
68	67	uneq2d 4123	. . . . . . . . . . . . . 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 4987	. . . . . . . . . . . . 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 4124	. . . . . . . . . . . 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 8428	. . . . . . . . . . 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 597	. . . . . . . . . 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 489	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘𝑛) ⊆ (wUniCl‘𝐴))
74	27	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → (wUniCl‘𝐴) ∈ WUni)
75	73	sselda 3938	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
76	74, 75	wunelss 10694	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → 𝑢 ⊆ (wUniCl‘𝐴))
77	76	ralrimiva 3157	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹‘𝑛)𝑢 ⊆ (wUniCl‘𝐴))
78		unissb 4907	. . . . . . . . . . . . 13 ⊢ (∪ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴) ↔ ∀𝑢 ∈ (𝐹‘𝑛)𝑢 ⊆ (wUniCl‘𝐴))
79	77, 78	sylibr 237	. . . . . . . . . . . 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 10693	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → 𝒫 𝑢 ∈ (wUniCl‘𝐴))
82	74, 75	wununi 10692	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → ∪ 𝑢 ∈ (wUniCl‘𝐴))
83	81, 82	prssd 4789	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → {𝒫 𝑢, ∪ 𝑢} ⊆ (wUniCl‘𝐴))
84	74	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → (wUniCl‘𝐴) ∈ WUni)
85	75	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
86		simplr 780	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → (𝐹‘𝑛) ⊆ (wUniCl‘𝐴))
87	86	sselda 3938	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → 𝑣 ∈ (wUniCl‘𝐴))
88	84, 85, 87	wunpr 10695	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → {𝑢, 𝑣} ∈ (wUniCl‘𝐴))
89	88	fmpttd 7112	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}):(𝐹‘𝑛)⟶(wUniCl‘𝐴))
90	89	frnd 6716	. . . . . . . . . . . . . 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 3157	. . . . . . . . . . . 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 5010	. . . . . . . . . . . 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 237	. . . . . . . . . . 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 3972	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))
97	96	ex 417	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
98	97	expcom 418	. . . . . . 7 ⊢ (𝑛 ∈ ω → (𝐴 ∈ 𝑉 → ((𝐹‘𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))))
99	13, 15, 17, 32, 98	finds2 7896	. . . . . 6 ⊢ (𝑚 ∈ ω → (𝐴 ∈ 𝑉 → (𝐹‘𝑚) ⊆ (wUniCl‘𝐴)))
100	99	com12 33	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑚 ∈ ω → (𝐹‘𝑚) ⊆ (wUniCl‘𝐴)))
101	100	ralrimiv 3156	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴))
102		iunss 5010	. . . 4 ⊢ (∪ 𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ ∀𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴))
103	101, 102	sylibr 237	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴))
104	11, 103	eqsstrid 3976	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑈 ⊆ (wUniCl‘𝐴))
105	5, 104	eqssd 3955	1 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 Vcvv 3455 ∪ cun 3904 ⊆ wss 3906 ∅c0 4287 𝒫 cpw 4563 {csn 4590 {cpr 4592 ∪ cuni 4873 ∪ ciun 4957 ↦ cmpt 5193 ran crn 5664 ↾ cres 5665 Oncon0 6362 suc csuc 6364 Fn wfn 6533 ‘cfv 6538 ωcom 7863 reccrdg 8397 1_oc1o 8447 WUnicwun 10686 wUniClcwunm 10687

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7415 df-om 7864 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-wun 10688 df-wunc 10689

This theorem is referenced by: (None)

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