Theorem r1ordg 9738

Description: Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.)

Assertion

Ref	Expression
r1ordg	⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵)))

Proof of Theorem r1ordg

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

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ dom 𝑅1)
2		r1funlim 9726	. . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
3	2	simpri 485	. . . . . . 7 ⊢ Lim dom 𝑅1
4		limord 6396	. . . . . . 7 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
5	3, 4	ax-mp 5	. . . . . 6 ⊢ Ord dom 𝑅1
6		ordsson 7762	. . . . . 6 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
7	5, 6	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 ⊆ On
8	7	sseli 3945	. . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On)
9	1, 8	syl 17	. . 3 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ On)
10		onelon 6360	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
11	8, 10	sylan 580	. . . 4 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
12		onsuc 7790	. . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
13	11, 12	syl 17	. . 3 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ∈ On)
14		eloni 6345	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
15		ordsucss 7796	. . . . . 6 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
16	14, 15	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
17	16	imp 406	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ 𝐵)
18	8, 17	sylan 580	. . 3 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ 𝐵)
19		eleq1 2817	. . . . . 6 ⊢ (𝑥 = suc 𝐴 → (𝑥 ∈ dom 𝑅1 ↔ suc 𝐴 ∈ dom 𝑅1))
20		fveq2 6861	. . . . . . 7 ⊢ (𝑥 = suc 𝐴 → (𝑅1‘𝑥) = (𝑅1‘suc 𝐴))
21	20	eleq2d 2815	. . . . . 6 ⊢ (𝑥 = suc 𝐴 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)))
22	19, 21	imbi12d 344	. . . . 5 ⊢ (𝑥 = suc 𝐴 → ((𝑥 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (suc 𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴))))
23		eleq1 2817	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1 ↔ 𝑦 ∈ dom 𝑅1))
24		fveq2 6861	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
25	24	eleq2d 2815	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈ (𝑅1‘𝑦)))
26	23, 25	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑦))))
27		eleq1 2817	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1))
28		fveq2 6861	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
29	28	eleq2d 2815	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝑦)))
30	27, 29	imbi12d 344	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝑦))))
31		eleq1 2817	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ dom 𝑅1 ↔ 𝐵 ∈ dom 𝑅1))
32		fveq2 6861	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑅1‘𝑥) = (𝑅1‘𝐵))
33	32	eleq2d 2815	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈ (𝑅1‘𝐵)))
34	31, 33	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (𝐵 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))))
35		fvex 6874	. . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V
36	35	pwid 4588	. . . . . . 7 ⊢ (𝑅1‘𝐴) ∈ 𝒫 (𝑅1‘𝐴)
37		limsuc 7828	. . . . . . . . 9 ⊢ (Lim dom 𝑅1 → (𝐴 ∈ dom 𝑅1 ↔ suc 𝐴 ∈ dom 𝑅1))
38	3, 37	ax-mp 5	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝑅1 ↔ suc 𝐴 ∈ dom 𝑅1)
39		r1sucg 9729	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
40	38, 39	sylbir 235	. . . . . . 7 ⊢ (suc 𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
41	36, 40	eleqtrrid 2836	. . . . . 6 ⊢ (suc 𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴))
42	41	a1i 11	. . . . 5 ⊢ (suc 𝐴 ∈ On → (suc 𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)))
43		limsuc 7828	. . . . . . . 8 ⊢ (Lim dom 𝑅1 → (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1))
44	3, 43	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1)
45		r1tr 9736	. . . . . . . . . . 11 ⊢ Tr (𝑅1‘𝑦)
46		dftr4 5224	. . . . . . . . . . 11 ⊢ (Tr (𝑅1‘𝑦) ↔ (𝑅1‘𝑦) ⊆ 𝒫 (𝑅1‘𝑦))
47	45, 46	mpbi 230	. . . . . . . . . 10 ⊢ (𝑅1‘𝑦) ⊆ 𝒫 (𝑅1‘𝑦)
48		r1sucg 9729	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
49	47, 48	sseqtrrid 3993	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝑅1 → (𝑅1‘𝑦) ⊆ (𝑅1‘suc 𝑦))
50	49	sseld 3948	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑅1 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑦) → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝑦)))
51	50	a2i 14	. . . . . . 7 ⊢ ((𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑦)) → (𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝑦)))
52	44, 51	biimtrrid 243	. . . . . 6 ⊢ ((𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑦)) → (suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝑦)))
53	52	a1i 11	. . . . 5 ⊢ (((𝑦 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑦) → ((𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑦)) → (suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝑦))))
54		simprl 770	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → suc 𝐴 ⊆ 𝑥)
55		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → suc 𝐴 ∈ On)
56		onsucb 7795	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
57	55, 56	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → 𝐴 ∈ On)
58		limord 6396	. . . . . . . . . . . . . 14 ⊢ (Lim 𝑥 → Ord 𝑥)
59	58	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → Ord 𝑥)
60		ordelsuc 7798	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
61	57, 59, 60	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
62	54, 61	mpbird 257	. . . . . . . . . . 11 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → 𝐴 ∈ 𝑥)
63		limsuc 7828	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
64	63	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
65	62, 64	mpbid 232	. . . . . . . . . 10 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → suc 𝐴 ∈ 𝑥)
66		simprr 772	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → 𝑥 ∈ dom 𝑅1)
67		ordtr1 6379	. . . . . . . . . . . . . 14 ⊢ (Ord dom 𝑅1 → ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → 𝐴 ∈ dom 𝑅1))
68	5, 67	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → 𝐴 ∈ dom 𝑅1)
69	62, 66, 68	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → 𝐴 ∈ dom 𝑅1)
70	69, 39	syl 17	. . . . . . . . . . 11 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
71	36, 70	eleqtrrid 2836	. . . . . . . . . 10 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴))
72		fveq2 6861	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝐴 → (𝑅1‘𝑦) = (𝑅1‘suc 𝐴))
73	72	eleq2d 2815	. . . . . . . . . . 11 ⊢ (𝑦 = suc 𝐴 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑦) ↔ (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)))
74	73	rspcev 3591	. . . . . . . . . 10 ⊢ ((suc 𝐴 ∈ 𝑥 ∧ (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)) → ∃𝑦 ∈ 𝑥 (𝑅1‘𝐴) ∈ (𝑅1‘𝑦))
75	65, 71, 74	syl2anc 584	. . . . . . . . 9 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → ∃𝑦 ∈ 𝑥 (𝑅1‘𝐴) ∈ (𝑅1‘𝑦))
76		eliun 4962	. . . . . . . . 9 ⊢ ((𝑅1‘𝐴) ∈ ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦) ↔ ∃𝑦 ∈ 𝑥 (𝑅1‘𝐴) ∈ (𝑅1‘𝑦))
77	75, 76	sylibr 234	. . . . . . . 8 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝑅1‘𝐴) ∈ ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
78		simpll 766	. . . . . . . . 9 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → Lim 𝑥)
79		r1limg 9731	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
80	66, 78, 79	syl2anc 584	. . . . . . . 8 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
81	77, 80	eleqtrrd 2832	. . . . . . 7 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥))
82	81	expr 456	. . . . . 6 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (𝑥 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥)))
83	82	a1d 25	. . . . 5 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (suc 𝐴 ⊆ 𝑦 → (𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑦))) → (𝑥 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥))))
84	22, 26, 30, 34, 42, 53, 83	tfindsg 7840	. . . 4 ⊢ (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝐵) → (𝐵 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵)))
85	84	impr 454	. . 3 ⊢ (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ dom 𝑅1)) → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))
86	9, 13, 18, 1, 85	syl22anc 838	. 2 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))
87	86	ex 412	1 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  𝒫 cpw 4566  ∪ ciun 4958  Tr wtr 5217  dom cdm 5641  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337  Fun wfun 6508  ‘cfv 6514  𝑅₁cr1 9722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-r1 9724

This theorem is referenced by:  r1ord3g  9739  r1ord  9740