Theorem rankonidlem 8574

Description: Lemma for rankonid 8575. (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 8512	. . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
2	1	simpri 477	. . . 4 ⊢ Lim dom 𝑅1
3		limord 5701	. . . 4 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
4	2, 3	ax-mp 5	. . 3 ⊢ Ord dom 𝑅1
5		ordelon 5664	. . 3 ⊢ ((Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → 𝐴 ∈ On)
6	4, 5	mpan 702	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
7		eleq1 2676	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1 ↔ 𝑦 ∈ dom 𝑅1))
8		eleq1 2676	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)))
9		fveq2 6103	. . . . . 6 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
10		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
11	9, 10	eqeq12d 2625	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦))
12	8, 11	anbi12d 743	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
13	7, 12	imbi12d 333	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))))
14		eleq1 2676	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅1 ↔ 𝐴 ∈ dom 𝑅1))
15		eleq1 2676	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ∈ ∪ (𝑅1 “ On)))
16		fveq2 6103	. . . . . 6 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
17		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
18	16, 17	eqeq12d 2625	. . . . 5 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴))
19	15, 18	anbi12d 743	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
20	14, 19	imbi12d 333	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))))
21		ordtr1 5684	. . . . . . . . . 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 468	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝑅1)
24		pm5.5 350	. . . . . . . 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 2968	. . . . . 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 788	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ 𝑥)
28		ordelon 5664	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Ord dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1) → 𝑥 ∈ On)
29	4, 28	mpan 702	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ dom 𝑅1 → 𝑥 ∈ On)
30	29	ad2antrr 758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
31		eloni 5650	. . . . . . . . . . . . . . . . . . . 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 6912	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑥 ∧ Ord 𝑥) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
34	27, 32, 33	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
35	27, 34	mpbid 221	. . . . . . . . . . . . . . . . 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 6941	. . . . . . . . . . . . . . . . . . . 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 207	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ∈ dom 𝑅1)
40		simpll 786	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom 𝑅1)
41		r1ord3g 8525	. . . . . . . . . . . . . . . . . 18 ⊢ ((suc 𝑦 ∈ dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1) → (suc 𝑦 ⊆ 𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥)))
42	39, 40, 41	syl2anc 691	. . . . . . . . . . . . . . . . 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 8546	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
45	44	ad2antrl 760	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
46		suceq 5707	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rank‘𝑦) = 𝑦 → suc (rank‘𝑦) = suc 𝑦)
47	46	ad2antll 761	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦)
48	47	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc 𝑦))
49	45, 48	eleqtrd 2690	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑦))
50	43, 49	sseldd 3569	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘𝑥))
51	50	ex 449	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → 𝑦 ∈ (𝑅1‘𝑥)))
52	51	ralimdva 2945	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥)))
53	52	imp 444	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥))
54		dfss3 3558	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥))
55	53, 54	sylibr 223	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ⊆ (𝑅1‘𝑥))
56		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
57	56	elpw 4114	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (𝑅1‘𝑥) ↔ 𝑥 ⊆ (𝑅1‘𝑥))
58	55, 57	sylibr 223	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ 𝒫 (𝑅1‘𝑥))
59		r1sucg 8515	. . . . . . . . . . 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 2691	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅1‘suc 𝑥))
62		r1elwf 8542	. . . . . . . . 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 8572	. . . . . . . . . 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 2676	. . . . . . . . . . . . . . . 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 2936	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
69		ralbi 3050	. . . . . . . . . . . . . 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 3558	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧)
72	70, 71	syl6bbr 277	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ 𝑥 ⊆ 𝑧))
73	72	rabbidv 3164	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
74	73	inteqd 4415	. . . . . . . . . 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 4432	. . . . . . . . . 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 2648	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = 𝑥)
80	63, 79	jca 553	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))
81	80	ex 449	. . . . . 6 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
82	26, 81	sylbid 229	. . . . 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 6949	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
86	6, 85	mpcom 37	1 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410  dom cdm 5038   “ cima 5041  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  Fun wfun 5798  ‘cfv 5804  𝑅₁cr1 8508  rankcrnk 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511

This theorem is referenced by:  rankonid  8575  onwf  8576  onssr1  8577