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Theorem wuncval2 9569

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 9560	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))
4		wuncss 9567	. . 3 ⊢ ((𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈) → (wUniCl‘𝐴) ⊆ 𝑈)
5	3, 4	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ⊆ 𝑈)
6		frfnom 7530	. . . . . 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 5985	. . . . . 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 221	. . . . 5 ⊢ 𝐹 Fn ω
9		fniunfv 6505	. . . . 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 2647	. . 3 ⊢ 𝑈 = ∪ 𝑚 ∈ ω (𝐹‘𝑚)
12		fveq2 6191	. . . . . . . 8 ⊢ (𝑚 = ∅ → (𝐹‘𝑚) = (𝐹‘∅))
13	12	sseq1d 3632	. . . . . . 7 ⊢ (𝑚 = ∅ → ((𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘∅) ⊆ (wUniCl‘𝐴)))
14		fveq2 6191	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
15	14	sseq1d 3632	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)))
16		fveq2 6191	. . . . . . . 8 ⊢ (𝑚 = suc 𝑛 → (𝐹‘𝑚) = (𝐹‘suc 𝑛))
17	16	sseq1d 3632	. . . . . . 7 ⊢ (𝑚 = suc 𝑛 → ((𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
18		1on 7567	. . . . . . . . . 10 ⊢ 1_𝑜 ∈ On
19		unexg 6959	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 1_𝑜 ∈ On) → (𝐴 ∪ 1_𝑜) ∈ V)
20	18, 19	mpan2 707	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 1_𝑜) ∈ V)
21	1	fveq1i 6192	. . . . . . . . . 10 ⊢ (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1_𝑜)) ↾ ω)‘∅)
22		fr0g 7531	. . . . . . . . . 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	syl5eq 2668	. . . . . . . . 9 ⊢ ((𝐴 ∪ 1_𝑜) ∈ V → (𝐹‘∅) = (𝐴 ∪ 1_𝑜))
24	20, 23	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘∅) = (𝐴 ∪ 1_𝑜))
25		wuncid 9565	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (wUniCl‘𝐴))
26		df1o2 7572	. . . . . . . . . 10 ⊢ 1_𝑜 = {∅}
27		wunccl 9566	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ∈ WUni)
28	27	wun0 9540	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ (wUniCl‘𝐴))
29	28	snssd 4340	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → {∅} ⊆ (wUniCl‘𝐴))
30	26, 29	syl5eqss 3649	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 1_𝑜 ⊆ (wUniCl‘𝐴))
31	25, 30	unssd 3789	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 1_𝑜) ⊆ (wUniCl‘𝐴))
32	24, 31	eqsstrd 3639	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘∅) ⊆ (wUniCl‘𝐴))
33		simplr 792	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → 𝑛 ∈ ω)
34		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑛) ∈ V
35	34	uniex 6953	. . . . . . . . . . . . 13 ⊢ ∪ (𝐹‘𝑛) ∈ V
36	34, 35	unex 6956	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ∈ V
37		prex 4909	. . . . . . . . . . . . . 14 ⊢ {𝒫 𝑢, ∪ 𝑢} ∈ V
38	34	mptex 6486	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}) ∈ V
39	38	rnex 7100	. . . . . . . . . . . . . 14 ⊢ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}) ∈ V
40	37, 39	unex 6956	. . . . . . . . . . . . 13 ⊢ ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ∈ V
41	34, 40	iunex 7147	. . . . . . . . . . . 12 ⊢ ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ∈ V
42	36, 41	unex 6956	. . . . . . . . . . 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 4444	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → ∪ 𝑤 = ∪ 𝑧)
45	43, 44	uneq12d 3768	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑤 ∪ ∪ 𝑤) = (𝑧 ∪ ∪ 𝑧))
46		pweq 4161	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥)
47		unieq 4444	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑥 → ∪ 𝑢 = ∪ 𝑥)
48	46, 47	preq12d 4276	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → {𝒫 𝑢, ∪ 𝑢} = {𝒫 𝑥, ∪ 𝑥})
49		preq1 4268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑥 → {𝑢, 𝑣} = {𝑥, 𝑣})
50	49	mpteq2dv 4745	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑥 → (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}))
51	50	rneqd 5353	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}))
52	48, 51	uneq12d 3768	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣})))
53	52	cbviunv 4559	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ∪ 𝑥 ∈ 𝑤 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}))
54		preq2 4269	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑦 → {𝑥, 𝑣} = {𝑥, 𝑦})
55	54	cbvmptv 4750	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}) = (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦})
56		mpteq1 4737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦}) = (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))
57	55, 56	syl5eq 2668	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}) = (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))
58	57	rneqd 5353	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣}) = ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))
59	58	uneq2d 3767	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑥, 𝑣})) = ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))
60	43, 59	iuneq12d 4546	. . . . . . . . . . . . . 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	syl5eq 2668	. . . . . . . . . . . . 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 3768	. . . . . . . . . . . 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 4444	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐹‘𝑛) → ∪ 𝑤 = ∪ (𝐹‘𝑛))
65	63, 64	uneq12d 3768	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐹‘𝑛) → (𝑤 ∪ ∪ 𝑤) = ((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)))
66		mpteq1 4737	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝐹‘𝑛) → (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))
67	66	rneqd 5353	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐹‘𝑛) → ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}))
68	67	uneq2d 3767	. . . . . . . . . . . . . 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 4546	. . . . . . . . . . . . 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 3768	. . . . . . . . . . . 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 7535	. . . . . . . . . . 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 694	. . . . . . . . . 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 477	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘𝑛) ⊆ (wUniCl‘𝐴))
74	27	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → (wUniCl‘𝐴) ∈ WUni)
75	73	sselda 3603	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
76	74, 75	wunelss 9530	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → 𝑢 ⊆ (wUniCl‘𝐴))
77	76	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹‘𝑛)𝑢 ⊆ (wUniCl‘𝐴))
78		unissb 4469	. . . . . . . . . . . . 13 ⊢ (∪ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴) ↔ ∀𝑢 ∈ (𝐹‘𝑛)𝑢 ⊆ (wUniCl‘𝐴))
79	77, 78	sylibr 224	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ∪ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴))
80	73, 79	unssd 3789	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ((𝐹‘𝑛) ∪ ∪ (𝐹‘𝑛)) ⊆ (wUniCl‘𝐴))
81	74, 75	wunpw 9529	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → 𝒫 𝑢 ∈ (wUniCl‘𝐴))
82	74, 75	wununi 9528	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → ∪ 𝑢 ∈ (wUniCl‘𝐴))
83		prssi 4353	. . . . . . . . . . . . . . 15 ⊢ ((𝒫 𝑢 ∈ (wUniCl‘𝐴) ∧ ∪ 𝑢 ∈ (wUniCl‘𝐴)) → {𝒫 𝑢, ∪ 𝑢} ⊆ (wUniCl‘𝐴))
84	81, 82, 83	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → {𝒫 𝑢, ∪ 𝑢} ⊆ (wUniCl‘𝐴))
85	74	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → (wUniCl‘𝐴) ∈ WUni)
86	75	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → 𝑢 ∈ (wUniCl‘𝐴))
87		simplr 792	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → (𝐹‘𝑛) ⊆ (wUniCl‘𝐴))
88	87	sselda 3603	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → 𝑣 ∈ (wUniCl‘𝐴))
89	85, 86, 88	wunpr 9531	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) ∧ 𝑣 ∈ (𝐹‘𝑛)) → {𝑢, 𝑣} ∈ (wUniCl‘𝐴))
90		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})
91	89, 90	fmptd 6385	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}):(𝐹‘𝑛)⟶(wUniCl‘𝐴))
92		frn 6053	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}):(𝐹‘𝑛)⟶(wUniCl‘𝐴) → ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}) ⊆ (wUniCl‘𝐴))
93	91, 92	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣}) ⊆ (wUniCl‘𝐴))
94	84, 93	unssd 3789	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) ∧ 𝑢 ∈ (𝐹‘𝑛)) → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
95	94	ralrimiva 2966	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ∀𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
96		iunss 4561	. . . . . . . . . . . 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))
97	95, 96	sylibr 224	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → ∪ 𝑢 ∈ (𝐹‘𝑛)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑛) ↦ {𝑢, 𝑣})) ⊆ (wUniCl‘𝐴))
98	80, 97	unssd 3789	. . . . . . . . . 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))
99	72, 98	eqsstrd 3639	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ⊆ (wUniCl‘𝐴)) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))
100	99	ex 450	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴)))
101	100	expcom 451	. . . . . . 7 ⊢ (𝑛 ∈ ω → (𝐴 ∈ 𝑉 → ((𝐹‘𝑛) ⊆ (wUniCl‘𝐴) → (𝐹‘suc 𝑛) ⊆ (wUniCl‘𝐴))))
102	13, 15, 17, 32, 101	finds2 7094	. . . . . 6 ⊢ (𝑚 ∈ ω → (𝐴 ∈ 𝑉 → (𝐹‘𝑚) ⊆ (wUniCl‘𝐴)))
103	102	com12 32	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑚 ∈ ω → (𝐹‘𝑚) ⊆ (wUniCl‘𝐴)))
104	103	ralrimiv 2965	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴))
105		iunss 4561	. . . 4 ⊢ (∪ 𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴) ↔ ∀𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴))
106	104, 105	sylibr 224	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑚 ∈ ω (𝐹‘𝑚) ⊆ (wUniCl‘𝐴))
107	11, 106	syl5eqss 3649	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑈 ⊆ (wUniCl‘𝐴))
108	5, 107	eqssd 3620	1 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 {cpr 4179 ∪ cuni 4436 ∪ ciun 4520 ↦ cmpt 4729 ran crn 5115 ↾ cres 5116 Oncon0 5723 suc csuc 5725 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 ωcom 7065 reccrdg 7505 1_𝑜c1o 7553 WUnicwun 9522 wUniClcwunm 9523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-wun 9524 df-wunc 9525

This theorem is referenced by: (None)

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