Theorem r1val1 9772

Description: The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
r1val1	⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem r1val1

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → 𝐴 = ∅)
2	1	fveq2d 6886	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) = (𝑅1‘∅))
3		r10 9754	. . . . 5 ⊢ (𝑅1‘∅) = ∅
4	2, 3	eqtrdi 2813	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) = ∅)
5		0ss 4353	. . . . 5 ⊢ ∅ ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
6	5	a1i 11	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → ∅ ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
7	4, 6	eqsstrd 3968	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
8		nfv 1947	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝑅1
9		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑥(𝑅1‘𝐴)
10		nfiu1 4990	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
11	9, 10	nfss 3927	. . . . 5 ⊢ Ⅎ𝑥(𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
12		simpr 490	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝐴 = suc 𝑥)
13	12	fveq2d 6886	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) = (𝑅1‘suc 𝑥))
14		eleq1 2850	. . . . . . . . . . . 12 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1))
15	14	biimpac 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
16		r1funlim 9752	. . . . . . . . . . . . 13 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
17	16	simpri 491	. . . . . . . . . . . 12 ⊢ Lim dom 𝑅1
18		limsuc 7849	. . . . . . . . . . . 12 ⊢ (Lim dom 𝑅1 → (𝑥 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1))
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1)
20	15, 19	sylibr 237	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝑥 ∈ dom 𝑅1)
21		r1sucg 9755	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
22	20, 21	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
23	13, 22	eqtrd 2797	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) = 𝒫 (𝑅1‘𝑥))
24		vex 3457	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
25	24	sucid 6446	. . . . . . . . . 10 ⊢ 𝑥 ∈ suc 𝑥
26	25, 12	eleqtrrid 2869	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝑥 ∈ 𝐴)
27		ssiun2 5010	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → 𝒫 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
28	26, 27	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝒫 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
29	23, 28	eqsstrd 3968	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
30	29	ex 418	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 = suc 𝑥 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)))
31	30	a1d 26	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑥 ∈ On → (𝐴 = suc 𝑥 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))))
32	8, 11, 31	rexlimd 3271	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)))
33	32	imp 412	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
34		r1limg 9757	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))
35		r1tr 9762	. . . . . . . . 9 ⊢ Tr (𝑅1‘𝑥)
36		dftr4 5222	. . . . . . . . 9 ⊢ (Tr (𝑅1‘𝑥) ↔ (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
37	35, 36	mpbi 233	. . . . . . . 8 ⊢ (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥)
38	37	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
39	38	ralrimivw 3160	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → ∀𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
40		ss2iun 4973	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥) → ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
41	39, 40	syl 18	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
42	34, 41	eqsstrd 3968	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
43	42	adantrl 729	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ (𝐴 ∈ V ∧ Lim 𝐴)) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
44		limord 6423	. . . . . . 7 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
45	17, 44	ax-mp 5	. . . . . 6 ⊢ Ord dom 𝑅1
46		ordsson 7786	. . . . . 6 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
47	45, 46	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 ⊆ On
48	47	sseli 3930	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
49		onzsl 7846	. . . 4 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
50	48, 49	sylib 221	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
51	7, 33, 43, 50	mpjao3dan 1459	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
52		ordtr1 6406	. . . . . . . 8 ⊢ (Ord dom 𝑅1 → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
53	45, 52	ax-mp 5	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
54	53	ancoms 464	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝑅1)
55	54, 21	syl 18	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
56		simpr 490	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
57		ordelord 6383	. . . . . . . . . 10 ⊢ ((Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → Ord 𝐴)
58	45, 57	mpan 703	. . . . . . . . 9 ⊢ (𝐴 ∈ dom 𝑅1 → Ord 𝐴)
59	58	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → Ord 𝐴)
60		ordelsuc 7820	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ Ord 𝐴) → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ⊆ 𝐴))
61	56, 59, 60	syl2anc 596	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ⊆ 𝐴))
62	56, 61	mpbid 235	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ 𝐴)
63	54, 19	sylib 221	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ∈ dom 𝑅1)
64		simpl 488	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ dom 𝑅1)
65		r1ord3g 9765	. . . . . . 7 ⊢ ((suc 𝑥 ∈ dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → (suc 𝑥 ⊆ 𝐴 → (𝑅1‘suc 𝑥) ⊆ (𝑅1‘𝐴)))
66	63, 64, 65	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (suc 𝑥 ⊆ 𝐴 → (𝑅1‘suc 𝑥) ⊆ (𝑅1‘𝐴)))
67	62, 66	mpd 16	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (𝑅1‘suc 𝑥) ⊆ (𝑅1‘𝐴))
68	55, 67	eqsstrrd 3969	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
69	68	ralrimiva 3156	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → ∀𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
70		iunss 5007	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴) ↔ ∀𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
71	69, 70	sylibr 237	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
72	51, 71	eqssd 3951	1 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560  ∪ ciun 4954  Tr wtr 5216  dom cdm 5659  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  Fun wfun 6531  ‘cfv 6537  𝑅₁cr1 9748

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-r1 9750

This theorem is used by:  rankr1ai  9784  r1val3  9824