Theorem rankonidlem 9713

Description: Lemma for rankonid 9714. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)

Assertion

Ref	Expression
rankonidlem	⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))

Proof of Theorem rankonidlem

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

Step	Hyp	Ref	Expression
1		r1funlim 9651	. . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
2	1	simpri 485	. . . 4 ⊢ Lim dom 𝑅1
3		limord 6363	. . . 4 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
4	2, 3	ax-mp 5	. . 3 ⊢ Ord dom 𝑅1
5		ordelon 6326	. . 3 ⊢ ((Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → 𝐴 ∈ On)
6	4, 5	mpan 690	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
7		eleq1 2817	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1 ↔ 𝑦 ∈ dom 𝑅1))
8		eleq1 2817	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)))
9		fveq2 6817	. . . . . 6 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
10		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
11	9, 10	eqeq12d 2746	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦))
12	8, 11	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
13	7, 12	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))))
14		eleq1 2817	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅1 ↔ 𝐴 ∈ dom 𝑅1))
15		eleq1 2817	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ∈ ∪ (𝑅1 “ On)))
16		fveq2 6817	. . . . . 6 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
17		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
18	16, 17	eqeq12d 2746	. . . . 5 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴))
19	15, 18	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
20	14, 19	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))))
21		ordtr1 6346	. . . . . . . . . 10 ⊢ (Ord dom 𝑅1 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → 𝑦 ∈ dom 𝑅1))
22	4, 21	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → 𝑦 ∈ dom 𝑅1)
23	22	ancoms 458	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝑅1)
24		pm5.5 361	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑅1 → ((𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
25	23, 24	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
26	25	ralbidva 3151	. . . . . 6 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
27		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ 𝑥)
28		ordelon 6326	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Ord dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1) → 𝑥 ∈ On)
29	4, 28	mpan 690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ dom 𝑅1 → 𝑥 ∈ On)
30	29	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
31		eloni 6312	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → Ord 𝑥)
32	30, 31	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → Ord 𝑥)
33		ordelsuc 7745	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑥 ∧ Ord 𝑥) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
34	27, 32, 33	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
35	27, 34	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ⊆ 𝑥)
36	23	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ dom 𝑅1)
37		limsuc 7774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim dom 𝑅1 → (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1))
38	2, 37	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1)
39	36, 38	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ∈ dom 𝑅1)
40		simpll 766	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom 𝑅1)
41		r1ord3g 9664	. . . . . . . . . . . . . . . . . 18 ⊢ ((suc 𝑦 ∈ dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1) → (suc 𝑦 ⊆ 𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥)))
42	39, 40, 41	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (suc 𝑦 ⊆ 𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥)))
43	35, 42	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥))
44		rankidb 9685	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
45	44	ad2antrl 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
46		suceq 6370	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rank‘𝑦) = 𝑦 → suc (rank‘𝑦) = suc 𝑦)
47	46	ad2antll 729	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦)
48	47	fveq2d 6821	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc 𝑦))
49	45, 48	eleqtrd 2831	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑦))
50	43, 49	sseldd 3933	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘𝑥))
51	50	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → 𝑦 ∈ (𝑅1‘𝑥)))
52	51	ralimdva 3142	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥)))
53	52	imp 406	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥))
54		dfss3 3921	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥))
55	53, 54	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ⊆ (𝑅1‘𝑥))
56		vex 3438	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
57	56	elpw 4552	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (𝑅1‘𝑥) ↔ 𝑥 ⊆ (𝑅1‘𝑥))
58	55, 57	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ 𝒫 (𝑅1‘𝑥))
59		r1sucg 9654	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
60	59	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
61	58, 60	eleqtrrd 2832	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅1‘suc 𝑥))
62		r1elwf 9681	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑅1‘suc 𝑥) → 𝑥 ∈ ∪ (𝑅1 “ On))
63	61, 62	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ ∪ (𝑅1 “ On))
64		rankval3b 9711	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (rank‘𝑥) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧})
65	63, 64	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧})
66		eleq1 2817	. . . . . . . . . . . . . . . 16 ⊢ ((rank‘𝑦) = 𝑦 → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
67	66	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
68	67	ralimi 3067	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
69		ralbi 3085	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧))
70	68, 69	syl 17	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧))
71		dfss3 3921	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧)
72	70, 71	bitr4di 289	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ 𝑥 ⊆ 𝑧))
73	72	rabbidv 3400	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
74	73	inteqd 4900	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
75	74	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
76	29	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
77		intmin 4916	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧} = 𝑥)
78	76, 77	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧} = 𝑥)
79	65, 75, 78	3eqtrd 2769	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = 𝑥)
80	63, 79	jca 511	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))
81	80	ex 412	. . . . . 6 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
82	26, 81	sylbid 240	. . . . 5 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
83	82	com12 32	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
84	83	a1i 11	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))))
85	13, 20, 84	tfis3 7783	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
86	6, 85	mpcom 38	1 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∀wral 3045  {crab 3393   ⊆ wss 3900  𝒫 cpw 4548  ∪ cuni 4857  ∩ cint 4895  dom cdm 5614   “ cima 5617  Ord word 6301  Oncon0 6302  Lim wlim 6303  suc csuc 6304  Fun wfun 6471  ‘cfv 6477  𝑅₁cr1 9647  rankcrnk 9648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  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 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-r1 9649  df-rank 9650

This theorem is referenced by:  rankonid  9714  onwf  9715  onssr1  9716