Theorem r1val1 9786

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 6887	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) = (𝑅1‘∅))
3		r10 9768	. . . . 5 ⊢ (𝑅1‘∅) = ∅
4	2, 3	eqtrdi 2812	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) = ∅)
5		0ss 4350	. . . . 5 ⊢ ∅ ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
6	5	a1i 11	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → ∅ ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
7	4, 6	eqsstrd 3965	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = ∅) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
8		nfv 1947	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝑅1
9		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥(𝑅1‘𝐴)
10		nfiu1 4986	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)
11	9, 10	nfss 3924	. . . . 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 6887	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) = (𝑅1‘suc 𝑥))
14		eleq1 2849	. . . . . . . . . . . 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		r1dmlim 9765	. . . . . . . . . . . 12 ⊢ Lim dom 𝑅1
17		limsuc 7858	. . . . . . . . . . . 12 ⊢ (Lim dom 𝑅1 → (𝑥 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1))
18	16, 17	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1)
19	15, 18	sylibr 237	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝑥 ∈ dom 𝑅1)
20		r1sucg 9769	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
21	19, 20	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
22	13, 21	eqtrd 2796	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) = 𝒫 (𝑅1‘𝑥))
23		vex 3455	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
24	23	sucid 6446	. . . . . . . . . 10 ⊢ 𝑥 ∈ suc 𝑥
25	24, 12	eleqtrrid 2868	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝑥 ∈ 𝐴)
26		ssiun2 5006	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → 𝒫 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
27	25, 26	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → 𝒫 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
28	22, 27	eqsstrd 3965	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐴 = suc 𝑥) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
29	28	ex 418	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 = suc 𝑥 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)))
30	29	a1d 26	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑥 ∈ On → (𝐴 = suc 𝑥 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))))
31	8, 11, 30	rexlimd 3270	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)))
32	31	imp 412	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
33		r1limg 9771	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))
34		r1tr 9776	. . . . . . . . 9 ⊢ Tr (𝑅1‘𝑥)
35		dftr4 5218	. . . . . . . . 9 ⊢ (Tr (𝑅1‘𝑥) ↔ (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
36	34, 35	mpbi 233	. . . . . . . 8 ⊢ (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥)
37	36	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
38	37	ralrimivw 3159	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → ∀𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥))
39		ss2iun 4970	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ 𝒫 (𝑅1‘𝑥) → ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
40	38, 39	syl 18	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
41	33, 40	eqsstrd 3965	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
42	41	adantrl 729	. . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ (𝐴 ∈ V ∧ Lim 𝐴)) → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
43		limord 6423	. . . . . . 7 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
44	16, 43	ax-mp 5	. . . . . 6 ⊢ Ord dom 𝑅1
45		ordsson 7795	. . . . . 6 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
46	44, 45	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 ⊆ On
47	46	sseli 3927	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
48		onzsl 7855	. . . 4 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
49	47, 48	sylib 221	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
50	7, 32, 42, 49	mpjao3dan 1459	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
51		ordtr1 6406	. . . . . . . 8 ⊢ (Ord dom 𝑅1 → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
52	44, 51	ax-mp 5	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
53	52	ancoms 464	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝑅1)
54	53, 20	syl 18	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
55		simpr 490	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
56		ordelord 6383	. . . . . . . . . 10 ⊢ ((Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → Ord 𝐴)
57	44, 56	mpan 703	. . . . . . . . 9 ⊢ (𝐴 ∈ dom 𝑅1 → Ord 𝐴)
58	57	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → Ord 𝐴)
59		ordelsuc 7829	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ Ord 𝐴) → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ⊆ 𝐴))
60	55, 58, 59	syl2anc 596	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ⊆ 𝐴))
61	55, 60	mpbid 235	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ 𝐴)
62	53, 18	sylib 221	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ∈ dom 𝑅1)
63		simpl 488	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ dom 𝑅1)
64		r1ord3g 9779	. . . . . . 7 ⊢ ((suc 𝑥 ∈ dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → (suc 𝑥 ⊆ 𝐴 → (𝑅1‘suc 𝑥) ⊆ (𝑅1‘𝐴)))
65	62, 63, 64	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (suc 𝑥 ⊆ 𝐴 → (𝑅1‘suc 𝑥) ⊆ (𝑅1‘𝐴)))
66	61, 65	mpd 16	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → (𝑅1‘suc 𝑥) ⊆ (𝑅1‘𝐴))
67	54, 66	eqsstrrd 3966	. . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴) → 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
68	67	ralrimiva 3155	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → ∀𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
69		iunss 5003	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴) ↔ ∀𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
70	68, 69	sylibr 237	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) ⊆ (𝑅1‘𝐴))
71	50, 70	eqssd 3948	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 3077  ∃wrex 3087  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ ciun 4951  Tr wtr 5212  dom cdm 5651  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  ‘cfv 6537  𝑅₁cr1 9759

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 7421  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-r1 9761

This theorem is used by:  rankr1ai  9799  r1val3  9843