Theorem rankonidlem 9744

Description: Lemma for rankonid 9745. (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 9682	. . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
2	1	simpri 485	. . . 4 ⊢ Lim dom 𝑅1
3		limord 6379	. . . 4 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
4	2, 3	ax-mp 5	. . 3 ⊢ Ord dom 𝑅1
5		ordelon 6342	. . 3 ⊢ ((Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → 𝐴 ∈ On)
6	4, 5	mpan 691	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
7		eleq1 2825	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1 ↔ 𝑦 ∈ dom 𝑅1))
8		eleq1 2825	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)))
9		fveq2 6835	. . . . . 6 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
10		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
11	9, 10	eqeq12d 2753	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦))
12	8, 11	anbi12d 633	. . . 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 2825	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅1 ↔ 𝐴 ∈ dom 𝑅1))
15		eleq1 2825	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ∈ ∪ (𝑅1 “ On)))
16		fveq2 6835	. . . . . 6 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
17		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
18	16, 17	eqeq12d 2753	. . . . 5 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴))
19	15, 18	anbi12d 633	. . . 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 6362	. . . . . . . . . 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 3158	. . . . . 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 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ 𝑥)
28		ordelon 6342	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Ord dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1) → 𝑥 ∈ On)
29	4, 28	mpan 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ dom 𝑅1 → 𝑥 ∈ On)
30	29	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
31		eloni 6328	. . . . . . . . . . . . . . . . . . . 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 7764	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑥 ∧ Ord 𝑥) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
34	27, 32, 33	syl2anc 585	. . . . . . . . . . . . . . . . . 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 7793	. . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom 𝑅1)
41		r1ord3g 9695	. . . . . . . . . . . . . . . . . 18 ⊢ ((suc 𝑦 ∈ dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1) → (suc 𝑦 ⊆ 𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥)))
42	39, 40, 41	syl2anc 585	. . . . . . . . . . . . . . . . 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 9716	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
45	44	ad2antrl 729	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
46		suceq 6386	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rank‘𝑦) = 𝑦 → suc (rank‘𝑦) = suc 𝑦)
47	46	ad2antll 730	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦)
48	47	fveq2d 6839	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc 𝑦))
49	45, 48	eleqtrd 2839	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑦))
50	43, 49	sseldd 3935	. . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . 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 3923	. . . . . . . . . . . 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 3445	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
57	56	elpw 4559	. . . . . . . . . . 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 9685	. . . . . . . . . . 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 2840	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅1‘suc 𝑥))
62		r1elwf 9712	. . . . . . . . 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 9742	. . . . . . . . . 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 2825	. . . . . . . . . . . . . . . 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 3074	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
69		ralbi 3092	. . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . 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 3407	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
74	73	inteqd 4908	. . . . . . . . . 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 4924	. . . . . . . . . 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 2776	. . . . . . . 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 7802	. 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 1542   ∈ wcel 2114  ∀wral 3052  {crab 3400   ⊆ wss 3902  𝒫 cpw 4555  ∪ cuni 4864  ∩ cint 4903  dom cdm 5625   “ cima 5628  Ord word 6317  Oncon0 6318  Lim wlim 6319  suc csuc 6320  Fun wfun 6487  ‘cfv 6493  𝑅₁cr1 9678  rankcrnk 9679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-r1 9680  df-rank 9681

This theorem is referenced by:  rankonid  9745  onwf  9746  onssr1  9747