Theorem r1ordg 8641

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 473	. . . 4 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ dom 𝑅1)
2		r1funlim 8629	. . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
3	2	simpri 478	. . . . . . 7 ⊢ Lim dom 𝑅1
4		limord 5784	. . . . . . 7 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
5	3, 4	ax-mp 5	. . . . . 6 ⊢ Ord dom 𝑅1
6		ordsson 6989	. . . . . 6 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
7	5, 6	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 ⊆ On
8	7	sseli 3599	. . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On)
9	1, 8	syl 17	. . 3 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ On)
10		onelon 5748	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
11	8, 10	sylan 488	. . . 4 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
12		suceloni 7013	. . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
13	11, 12	syl 17	. . 3 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ∈ On)
14		eloni 5733	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
15		ordsucss 7018	. . . . . 6 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
16	14, 15	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
17	16	imp 445	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ 𝐵)
18	8, 17	sylan 488	. . 3 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ 𝐵)
19		eleq1 2689	. . . . . 6 ⊢ (𝑥 = suc 𝐴 → (𝑥 ∈ dom 𝑅1 ↔ suc 𝐴 ∈ dom 𝑅1))
20		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = suc 𝐴 → (𝑅1‘𝑥) = (𝑅1‘suc 𝐴))
21	20	eleq2d 2687	. . . . . 6 ⊢ (𝑥 = suc 𝐴 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)))
22	19, 21	imbi12d 334	. . . . 5 ⊢ (𝑥 = suc 𝐴 → ((𝑥 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (suc 𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴))))
23		eleq1 2689	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1 ↔ 𝑦 ∈ dom 𝑅1))
24		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
25	24	eleq2d 2687	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈ (𝑅1‘𝑦)))
26	23, 25	imbi12d 334	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑦))))
27		eleq1 2689	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1))
28		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
29	28	eleq2d 2687	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝑦)))
30	27, 29	imbi12d 334	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝑦))))
31		eleq1 2689	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ dom 𝑅1 ↔ 𝐵 ∈ dom 𝑅1))
32		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑅1‘𝑥) = (𝑅1‘𝐵))
33	32	eleq2d 2687	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈ (𝑅1‘𝐵)))
34	31, 33	imbi12d 334	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (𝐵 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))))
35		fvex 6201	. . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V
36	35	pwid 4174	. . . . . . 7 ⊢ (𝑅1‘𝐴) ∈ 𝒫 (𝑅1‘𝐴)
37		limsuc 7049	. . . . . . . . 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 8632	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
40	38, 39	sylbir 225	. . . . . . 7 ⊢ (suc 𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
41	36, 40	syl5eleqr 2708	. . . . . 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 7049	. . . . . . . 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 8639	. . . . . . . . . . 11 ⊢ Tr (𝑅1‘𝑦)
46		dftr4 4757	. . . . . . . . . . 11 ⊢ (Tr (𝑅1‘𝑦) ↔ (𝑅1‘𝑦) ⊆ 𝒫 (𝑅1‘𝑦))
47	45, 46	mpbi 220	. . . . . . . . . 10 ⊢ (𝑅1‘𝑦) ⊆ 𝒫 (𝑅1‘𝑦)
48		r1sucg 8632	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
49	47, 48	syl5sseqr 3654	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝑅1 → (𝑅1‘𝑦) ⊆ (𝑅1‘suc 𝑦))
50	49	sseld 3602	. . . . . . . 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	syl5bir 233	. . . . . 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 794	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → suc 𝐴 ⊆ 𝑥)
55		simplr 792	. . . . . . . . . . . . . 14 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → suc 𝐴 ∈ On)
56		sucelon 7017	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
57	55, 56	sylibr 224	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → 𝐴 ∈ On)
58		limord 5784	. . . . . . . . . . . . . 14 ⊢ (Lim 𝑥 → Ord 𝑥)
59	58	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → Ord 𝑥)
60		ordelsuc 7020	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
61	57, 59, 60	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
62	54, 61	mpbird 247	. . . . . . . . . . 11 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → 𝐴 ∈ 𝑥)
63		limsuc 7049	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
64	63	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
65	62, 64	mpbid 222	. . . . . . . . . 10 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → suc 𝐴 ∈ 𝑥)
66		simprr 796	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → 𝑥 ∈ dom 𝑅1)
67		ordtr1 5767	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . 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	syl5eleqr 2708	. . . . . . . . . 10 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴))
72		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝐴 → (𝑅1‘𝑦) = (𝑅1‘suc 𝐴))
73	72	eleq2d 2687	. . . . . . . . . . 11 ⊢ (𝑦 = suc 𝐴 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑦) ↔ (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)))
74	73	rspcev 3309	. . . . . . . . . 10 ⊢ ((suc 𝐴 ∈ 𝑥 ∧ (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)) → ∃𝑦 ∈ 𝑥 (𝑅1‘𝐴) ∈ (𝑅1‘𝑦))
75	65, 71, 74	syl2anc 693	. . . . . . . . 9 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → ∃𝑦 ∈ 𝑥 (𝑅1‘𝐴) ∈ (𝑅1‘𝑦))
76		eliun 4524	. . . . . . . . 9 ⊢ ((𝑅1‘𝐴) ∈ ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦) ↔ ∃𝑦 ∈ 𝑥 (𝑅1‘𝐴) ∈ (𝑅1‘𝑦))
77	75, 76	sylibr 224	. . . . . . . 8 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝑅1‘𝐴) ∈ ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
78		simpll 790	. . . . . . . . 9 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → Lim 𝑥)
79		r1limg 8634	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
80	66, 78, 79	syl2anc 693	. . . . . . . 8 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
81	77, 80	eleqtrrd 2704	. . . . . . 7 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) → (𝑅1‘𝐴) ∈ (𝑅1‘𝑥))
82	81	expr 643	. . . . . 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 7060	. . . 4 ⊢ (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝐵) → (𝐵 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵)))
85	84	impr 649	. . 3 ⊢ (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ dom 𝑅1)) → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))
86	9, 13, 18, 1, 85	syl22anc 1327	. 2 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵) → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))
87	86	ex 450	1 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  𝒫 cpw 4158  ∪ ciun 4520  Tr wtr 4752  dom cdm 5114  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  Fun wfun 5882  ‘cfv 5888  𝑅₁cr1 8625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627

This theorem is referenced by:  r1ord3g  8642  r1ord  8643