Theorem r1val1 9777

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 485	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → 𝐴 = ∅)
2	1	fveq2d 6892	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) = (𝑅1‘∅))
3		r10 9759	. . . . 5 ⊢ (𝑅1‘∅) = ∅
4	2, 3	eqtrdi 2788	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) = ∅)
5		0ss 4395	. . . . 5 ⊢ ∅ ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
6	5	a1i 11	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → ∅ ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
7	4, 6	eqsstrd 4019	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
8		nfv 1917	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝑅1
9		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑥(𝑅1‘𝐴)
10		nfiu1 5030	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
11	9, 10	nfss 3973	. . . . 5 ⊢ Ⅎ𝑥(𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
12		simpr 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝐴 = suc 𝑥)
13	12	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) = (𝑅1‘suc 𝑥))
14		eleq1 2821	. . . . . . . . . . . 12 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1))
15	14	biimpac 479	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
16		r1funlim 9757	. . . . . . . . . . . . 13 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
17	16	simpri 486	. . . . . . . . . . . 12 ⊢ Lim dom 𝑅1
18		limsuc 7834	. . . . . . . . . . . 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 233	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝑥 ∈ dom 𝑅1)
21		r1sucg 9760	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
22	20, 21	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
23	13, 22	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) = 𝒫 (𝑅1‘𝑥))
24		vex 3478	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
25	24	sucid 6443	. . . . . . . . . 10 ⊢ 𝑥 ∈ suc 𝑥
26	25, 12	eleqtrrid 2840	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝑥 ∈ 𝐴)
27		ssiun2 5049	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → 𝒫 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
28	26, 27	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝒫 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
29	23, 28	eqsstrd 4019	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
30	29	ex 413	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 = suc 𝑥 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)))
31	30	a1d 25	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑥 ∈ On → (𝐴 = suc 𝑥 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))))
32	8, 11, 31	rexlimd 3263	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)))
33	32	imp 407	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
34		r1limg 9762	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))
35		r1tr 9767	. . . . . . . . 9 ⊢ Tr (𝑅1‘𝑥)
36		dftr4 5271	. . . . . . . . 9 ⊢ (Tr (𝑅1‘𝑥) ↔ (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
37	35, 36	mpbi 229	. . . . . . . 8 ⊢ (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥)
38	37	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
39	38	ralrimivw 3150	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → ∀𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
40		ss2iun 5014	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥) → ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
41	39, 40	syl 17	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
42	34, 41	eqsstrd 4019	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
43	42	adantrl 714	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ (𝐴 ∈ V ∧ Lim 𝐴)) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
44		limord 6421	. . . . . . 7 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
45	17, 44	ax-mp 5	. . . . . 6 ⊢ Ord dom 𝑅1
46		ordsson 7766	. . . . . 6 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
47	45, 46	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 ⊆ On
48	47	sseli 3977	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
49		onzsl 7831	. . . 4 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
50	48, 49	sylib 217	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
51	7, 33, 43, 50	mpjao3dan 1431	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
52		ordtr1 6404	. . . . . . . 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 459	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝑅1)
55	54, 21	syl 17	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
56		simpr 485	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
57		ordelord 6383	. . . . . . . . . 10 ⊢ ((Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → Ord 𝐴)
58	45, 57	mpan 688	. . . . . . . . 9 ⊢ (𝐴 ∈ dom 𝑅1 → Ord 𝐴)
59	58	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → Ord 𝐴)
60		ordelsuc 7804	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ Ord 𝐴) → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ⊆ 𝐴))
61	56, 59, 60	syl2anc 584	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ⊆ 𝐴))
62	56, 61	mpbid 231	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ 𝐴)
63	54, 19	sylib 217	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ∈ dom 𝑅1)
64		simpl 483	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ dom 𝑅1)
65		r1ord3g 9770	. . . . . . 7 ⊢ ((suc 𝑥 ∈ dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → (suc 𝑥 ⊆ 𝐴 → (𝑅1‘suc 𝑥) ⊆ (𝑅1‘𝐴)))
66	63, 64, 65	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (suc 𝑥 ⊆ 𝐴 → (𝑅1‘suc 𝑥) ⊆ (𝑅1‘𝐴)))
67	62, 66	mpd 15	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (𝑅1‘suc 𝑥) ⊆ (𝑅1‘𝐴))
68	55, 67	eqsstrrd 4020	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
69	68	ralrimiva 3146	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → ∀𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
70		iunss 5047	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴) ↔ ∀𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
71	69, 70	sylibr 233	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
72	51, 71	eqssd 3998	1 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  ∪ ciun 4996  Tr wtr 5264  dom cdm 5675  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  Fun wfun 6534  ‘cfv 6540  𝑅₁cr1 9753

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-r1 9755

This theorem is referenced by:  rankr1ai  9789  r1val3  9829