Theorem r1val1 8933

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 479	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → 𝐴 = ∅)
2	1	fveq2d 6441	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) = (𝑅1‘∅))
3		r10 8915	. . . . 5 ⊢ (𝑅1‘∅) = ∅
4	2, 3	syl6eq 2877	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) = ∅)
5		0ss 4199	. . . . 5 ⊢ ∅ ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
6	5	a1i 11	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → ∅ ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
7	4, 6	eqsstrd 3864	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
8		nfv 2013	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝑅1
9		nfcv 2969	. . . . . 6 ⊢ Ⅎ𝑥(𝑅1‘𝐴)
10		nfiu1 4772	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
11	9, 10	nfss 3820	. . . . 5 ⊢ Ⅎ𝑥(𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
12		simpr 479	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝐴 = suc 𝑥)
13	12	fveq2d 6441	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) = (𝑅1‘suc 𝑥))
14		eleq1 2894	. . . . . . . . . . . 12 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1))
15	14	biimpac 472	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
16		r1funlim 8913	. . . . . . . . . . . . 13 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
17	16	simpri 481	. . . . . . . . . . . 12 ⊢ Lim dom 𝑅1
18		limsuc 7315	. . . . . . . . . . . 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 226	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝑥 ∈ dom 𝑅1)
21		r1sucg 8916	. . . . . . . . . 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 2861	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) = 𝒫 (𝑅1‘𝑥))
24		vex 3417	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
25	24	sucid 6046	. . . . . . . . . 10 ⊢ 𝑥 ∈ suc 𝑥
26	25, 12	syl5eleqr 2913	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝑥 ∈ 𝐴)
27		ssiun2 4785	. . . . . . . . 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 3864	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
30	29	ex 403	. . . . . 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 3235	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)))
33	32	imp 397	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
34		r1limg 8918	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))
35		r1tr 8923	. . . . . . . . 9 ⊢ Tr (𝑅1‘𝑥)
36		dftr4 4982	. . . . . . . . 9 ⊢ (Tr (𝑅1‘𝑥) ↔ (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
37	35, 36	mpbi 222	. . . . . . . 8 ⊢ (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥)
38	37	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
39	38	ralrimivw 3176	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → ∀𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
40		ss2iun 4758	. . . . . 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 3864	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
43	42	adantrl 707	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ (𝐴 ∈ V ∧ Lim 𝐴)) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
44		limord 6026	. . . . . . 7 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
45	17, 44	ax-mp 5	. . . . . 6 ⊢ Ord dom 𝑅1
46		ordsson 7255	. . . . . 6 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
47	45, 46	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 ⊆ On
48	47	sseli 3823	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
49		onzsl 7312	. . . 4 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
50	48, 49	sylib 210	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
51	7, 33, 43, 50	mpjao3dan 1560	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
52		ordtr1 6010	. . . . . . . 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 452	. . . . . 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 479	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
57		ordelord 5989	. . . . . . . . . 10 ⊢ ((Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → Ord 𝐴)
58	45, 57	mpan 681	. . . . . . . . 9 ⊢ (𝐴 ∈ dom 𝑅1 → Ord 𝐴)
59	58	adantr 474	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → Ord 𝐴)
60		ordelsuc 7286	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ Ord 𝐴) → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ⊆ 𝐴))
61	56, 59, 60	syl2anc 579	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ⊆ 𝐴))
62	56, 61	mpbid 224	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ 𝐴)
63	54, 19	sylib 210	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ∈ dom 𝑅1)
64		simpl 476	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ dom 𝑅1)
65		r1ord3g 8926	. . . . . . 7 ⊢ ((suc 𝑥 ∈ dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → (suc 𝑥 ⊆ 𝐴 → (𝑅1‘suc 𝑥) ⊆ (𝑅1‘𝐴)))
66	63, 64, 65	syl2anc 579	. . . . . 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	eqsstr3d 3865	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
69	68	ralrimiva 3175	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → ∀𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
70		iunss 4783	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴) ↔ ∀𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
71	69, 70	sylibr 226	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
72	51, 71	eqssd 3844	1 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∨ w3o 1110   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118  Vcvv 3414   ⊆ wss 3798  ∅c0 4146  𝒫 cpw 4380  ∪ ciun 4742  Tr wtr 4977  dom cdm 5346  Ord word 5966  Oncon0 5967  Lim wlim 5968  suc csuc 5969  Fun wfun 6121  ‘cfv 6127  𝑅₁cr1 8909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-om 7332  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-r1 8911

This theorem is referenced by:  rankr1ai  8945  r1val3  8985