Theorem r1pwss 9064

Description: Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
r1pwss	⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵))

Proof of Theorem r1pwss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1funlim 9046	. . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
2	1	simpri 486	. . . . . 6 ⊢ Lim dom 𝑅1
3		limord 6130	. . . . . 6 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
4	2, 3	ax-mp 5	. . . . 5 ⊢ Ord dom 𝑅1
5		ordsson 7365	. . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
6	4, 5	ax-mp 5	. . . 4 ⊢ dom 𝑅1 ⊆ On
7		elfvdm 6575	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
8	6, 7	sseldi 3891	. . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On)
9		onzsl 7422	. . 3 ⊢ (𝐵 ∈ On ↔ (𝐵 = ∅ ∨ ∃𝑥 ∈ On 𝐵 = suc 𝑥 ∨ (𝐵 ∈ V ∧ Lim 𝐵)))
10	8, 9	sylib 219	. 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝐵 = ∅ ∨ ∃𝑥 ∈ On 𝐵 = suc 𝑥 ∨ (𝐵 ∈ V ∧ Lim 𝐵)))
11		noel 4220	. . . . 5 ⊢ ¬ 𝐴 ∈ ∅
12		fveq2 6543	. . . . . . . 8 ⊢ (𝐵 = ∅ → (𝑅1‘𝐵) = (𝑅1‘∅))
13		r10 9048	. . . . . . . 8 ⊢ (𝑅1‘∅) = ∅
14	12, 13	syl6eq 2847	. . . . . . 7 ⊢ (𝐵 = ∅ → (𝑅1‘𝐵) = ∅)
15	14	eleq2d 2868	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐴 ∈ ∅))
16	15	biimpcd 250	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝐵 = ∅ → 𝐴 ∈ ∅))
17	11, 16	mtoi 200	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ¬ 𝐵 = ∅)
18	17	pm2.21d 121	. . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝐵 = ∅ → 𝒫 𝐴 ⊆ (𝑅1‘𝐵)))
19		simpl 483	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → 𝐴 ∈ (𝑅1‘𝐵))
20		simpr 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → 𝐵 = suc 𝑥)
21	20	fveq2d 6547	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → (𝑅1‘𝐵) = (𝑅1‘suc 𝑥))
22	7	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → 𝐵 ∈ dom 𝑅1)
23	20, 22	eqeltrrd 2884	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
24		limsuc 7425	. . . . . . . . . . . 12 ⊢ (Lim dom 𝑅1 → (𝑥 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1))
25	2, 24	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1)
26	23, 25	sylibr 235	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → 𝑥 ∈ dom 𝑅1)
27		r1sucg 9049	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
28	26, 27	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
29	21, 28	eqtrd 2831	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → (𝑅1‘𝐵) = 𝒫 (𝑅1‘𝑥))
30	19, 29	eleqtrd 2885	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → 𝐴 ∈ 𝒫 (𝑅1‘𝑥))
31		elpwi 4467	. . . . . . . 8 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘𝑥) → 𝐴 ⊆ (𝑅1‘𝑥))
32		sspwb 5238	. . . . . . . 8 ⊢ (𝐴 ⊆ (𝑅1‘𝑥) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘𝑥))
33	31, 32	sylib 219	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘𝑥) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘𝑥))
34	30, 33	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘𝑥))
35	34, 29	sseqtr4d 3933	. . . . 5 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐵 = suc 𝑥) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵))
36	35	ex 413	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝐵 = suc 𝑥 → 𝒫 𝐴 ⊆ (𝑅1‘𝐵)))
37	36	rexlimdvw 3253	. . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (∃𝑥 ∈ On 𝐵 = suc 𝑥 → 𝒫 𝐴 ⊆ (𝑅1‘𝐵)))
38		r1tr 9056	. . . . . 6 ⊢ Tr (𝑅1‘𝐵)
39		simpl 483	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) → 𝐴 ∈ (𝑅1‘𝐵))
40		r1limg 9051	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ Lim 𝐵) → (𝑅1‘𝐵) = ∪ 𝑥 ∈ 𝐵 (𝑅1‘𝑥))
41	7, 40	sylan 580	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) → (𝑅1‘𝐵) = ∪ 𝑥 ∈ 𝐵 (𝑅1‘𝑥))
42	39, 41	eleqtrd 2885	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 (𝑅1‘𝑥))
43		eliun 4833	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 (𝑅1‘𝑥) ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ (𝑅1‘𝑥))
44	42, 43	sylib 219	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) → ∃𝑥 ∈ 𝐵 𝐴 ∈ (𝑅1‘𝑥))
45		simprl 767	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → 𝑥 ∈ 𝐵)
46		limsuc 7425	. . . . . . . . . . . . 13 ⊢ (Lim 𝐵 → (𝑥 ∈ 𝐵 ↔ suc 𝑥 ∈ 𝐵))
47	46	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → (𝑥 ∈ 𝐵 ↔ suc 𝑥 ∈ 𝐵))
48	45, 47	mpbid 233	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → suc 𝑥 ∈ 𝐵)
49		limsuc 7425	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → (suc 𝑥 ∈ 𝐵 ↔ suc suc 𝑥 ∈ 𝐵))
50	49	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → (suc 𝑥 ∈ 𝐵 ↔ suc suc 𝑥 ∈ 𝐵))
51	48, 50	mpbid 233	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → suc suc 𝑥 ∈ 𝐵)
52		r1tr 9056	. . . . . . . . . . . . . . 15 ⊢ Tr (𝑅1‘𝑥)
53		simprr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → 𝐴 ∈ (𝑅1‘𝑥))
54		trss 5077	. . . . . . . . . . . . . . 15 ⊢ (Tr (𝑅1‘𝑥) → (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ⊆ (𝑅1‘𝑥)))
55	52, 53, 54	mpsyl 68	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → 𝐴 ⊆ (𝑅1‘𝑥))
56	55, 32	sylib 219	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘𝑥))
57	7	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → 𝐵 ∈ dom 𝑅1)
58		ordtr1 6114	. . . . . . . . . . . . . . . 16 ⊢ (Ord dom 𝑅1 → ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
59	4, 58	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
60	45, 57, 59	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → 𝑥 ∈ dom 𝑅1)
61	60, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
62	56, 61	sseqtr4d 3933	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → 𝒫 𝐴 ⊆ (𝑅1‘suc 𝑥))
63		fvex 6556	. . . . . . . . . . . . 13 ⊢ (𝑅1‘suc 𝑥) ∈ V
64	63	elpw2 5144	. . . . . . . . . . . 12 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc 𝑥) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc 𝑥))
65	62, 64	sylibr 235	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc 𝑥))
66	60, 25	sylib 219	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → suc 𝑥 ∈ dom 𝑅1)
67		r1sucg 9049	. . . . . . . . . . . 12 ⊢ (suc 𝑥 ∈ dom 𝑅1 → (𝑅1‘suc suc 𝑥) = 𝒫 (𝑅1‘suc 𝑥))
68	66, 67	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → (𝑅1‘suc suc 𝑥) = 𝒫 (𝑅1‘suc 𝑥))
69	65, 68	eleqtrrd 2886	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥))
70		fveq2 6543	. . . . . . . . . . . 12 ⊢ (𝑦 = suc suc 𝑥 → (𝑅1‘𝑦) = (𝑅1‘suc suc 𝑥))
71	70	eleq2d 2868	. . . . . . . . . . 11 ⊢ (𝑦 = suc suc 𝑥 → (𝒫 𝐴 ∈ (𝑅1‘𝑦) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥)))
72	71	rspcev 3559	. . . . . . . . . 10 ⊢ ((suc suc 𝑥 ∈ 𝐵 ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥)) → ∃𝑦 ∈ 𝐵 𝒫 𝐴 ∈ (𝑅1‘𝑦))
73	51, 69, 72	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ (𝑅1‘𝑥))) → ∃𝑦 ∈ 𝐵 𝒫 𝐴 ∈ (𝑅1‘𝑦))
74	44, 73	rexlimddv 3254	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) → ∃𝑦 ∈ 𝐵 𝒫 𝐴 ∈ (𝑅1‘𝑦))
75		eliun 4833	. . . . . . . 8 ⊢ (𝒫 𝐴 ∈ ∪ 𝑦 ∈ 𝐵 (𝑅1‘𝑦) ↔ ∃𝑦 ∈ 𝐵 𝒫 𝐴 ∈ (𝑅1‘𝑦))
76	74, 75	sylibr 235	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 ∈ ∪ 𝑦 ∈ 𝐵 (𝑅1‘𝑦))
77		r1limg 9051	. . . . . . . 8 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ Lim 𝐵) → (𝑅1‘𝐵) = ∪ 𝑦 ∈ 𝐵 (𝑅1‘𝑦))
78	7, 77	sylan 580	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) → (𝑅1‘𝐵) = ∪ 𝑦 ∈ 𝐵 (𝑅1‘𝑦))
79	76, 78	eleqtrrd 2886	. . . . . 6 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 ∈ (𝑅1‘𝐵))
80		trss 5077	. . . . . 6 ⊢ (Tr (𝑅1‘𝐵) → (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵)))
81	38, 79, 80	mpsyl 68	. . . . 5 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵))
82	81	ex 413	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (Lim 𝐵 → 𝒫 𝐴 ⊆ (𝑅1‘𝐵)))
83	82	adantld 491	. . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ((𝐵 ∈ V ∧ Lim 𝐵) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵)))
84	18, 37, 83	3jaod 1421	. 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ((𝐵 = ∅ ∨ ∃𝑥 ∈ On 𝐵 = suc 𝑥 ∨ (𝐵 ∈ V ∧ Lim 𝐵)) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵)))
85	10, 84	mpd 15	1 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ w3o 1079   = wceq 1522   ∈ wcel 2081  ∃wrex 3106  Vcvv 3437   ⊆ wss 3863  ∅c0 4215  𝒫 cpw 4457  ∪ ciun 4829  Tr wtr 5068  dom cdm 5448  Ord word 6070  Oncon0 6071  Lim wlim 6072  suc csuc 6073  Fun wfun 6224  ‘cfv 6230  𝑅₁cr1 9042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-om 7442  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-r1 9044

This theorem is referenced by:  r1sscl  9065