Theorem rankonidlem 9799

Description: Lemma for rankonid 9800. (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 9737	. . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
2	1	simpri 490	. . . 4 ⊢ Lim dom 𝑅1
3		limord 6422	. . . 4 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
4	2, 3	ax-mp 5	. . 3 ⊢ Ord dom 𝑅1
5		ordelon 6384	. . 3 ⊢ ((Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → 𝐴 ∈ On)
6	4, 5	mpan 702	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
7		eleq1 2849	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1 ↔ 𝑦 ∈ dom 𝑅1))
8		eleq1 2849	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)))
9		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
10		id 23	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
11	9, 10	eqeq12d 2777	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦))
12	8, 11	anbi12d 643	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
13	7, 12	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))))
14		eleq1 2849	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅1 ↔ 𝐴 ∈ dom 𝑅1))
15		eleq1 2849	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ∈ ∪ (𝑅1 “ On)))
16		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
17		id 23	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
18	16, 17	eqeq12d 2777	. . . . 5 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴))
19	15, 18	anbi12d 643	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
20	14, 19	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))))
21		ordtr1 6405	. . . . . . . . . 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 463	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝑅1)
24		pm5.5 364	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑅1 → ((𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
25	23, 24	syl 18	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
26	25	ralbidva 3184	. . . . . 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 780	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ 𝑥)
28		ordelon 6384	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Ord dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1) → 𝑥 ∈ On)
29	4, 28	mpan 702	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ dom 𝑅1 → 𝑥 ∈ On)
30	29	ad2antrr 738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
31		eloni 6370	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → Ord 𝑥)
32	30, 31	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → Ord 𝑥)
33		ordelsuc 7815	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑥 ∧ Ord 𝑥) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
34	27, 32, 33	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
35	27, 34	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ⊆ 𝑥)
36	23	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ dom 𝑅1)
37		limsuc 7844	. . . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ∈ dom 𝑅1)
40		simpll 778	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom 𝑅1)
41		r1ord3g 9750	. . . . . . . . . . . . . . . . . 18 ⊢ ((suc 𝑦 ∈ dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1) → (suc 𝑦 ⊆ 𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥)))
42	39, 40, 41	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (suc 𝑦 ⊆ 𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥)))
43	35, 42	mpd 16	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥))
44		rankidb 9771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
45	44	ad2antrl 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
46		suceq 6429	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rank‘𝑦) = 𝑦 → suc (rank‘𝑦) = suc 𝑦)
47	46	ad2antll 741	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦)
48	47	fveq2d 6885	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc 𝑦))
49	45, 48	eleqtrd 2863	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑦))
50	43, 49	sseldd 3937	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘𝑥))
51	50	ex 417	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → 𝑦 ∈ (𝑅1‘𝑥)))
52	51	ralimdva 3175	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥)))
53	52	imp 411	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥))
54		dfss3 3925	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥))
55	53, 54	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ⊆ (𝑅1‘𝑥))
56		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
57	56	elpw 4565	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (𝑅1‘𝑥) ↔ 𝑥 ⊆ (𝑅1‘𝑥))
58	55, 57	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ 𝒫 (𝑅1‘𝑥))
59		r1sucg 9740	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
60	59	adantr 485	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
61	58, 60	eleqtrrd 2864	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅1‘suc 𝑥))
62		r1elwf 9767	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑅1‘suc 𝑥) → 𝑥 ∈ ∪ (𝑅1 “ On))
63	61, 62	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ ∪ (𝑅1 “ On))
64		rankval3b 9797	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (rank‘𝑥) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧})
65	63, 64	syl 18	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧})
66		eleq1 2849	. . . . . . . . . . . . . . . 16 ⊢ ((rank‘𝑦) = 𝑦 → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
67	66	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
68	67	ralimi 3100	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
69		ralbi 3118	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧))
70	68, 69	syl 18	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧))
71		dfss3 3925	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧)
72	70, 71	bitr4di 292	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ 𝑥 ⊆ 𝑧))
73	72	rabbidv 3421	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
74	73	inteqd 4916	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
75	74	adantl 486	. . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
77		intmin 4932	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧} = 𝑥)
78	76, 77	syl 18	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧} = 𝑥)
79	65, 75, 78	3eqtrd 2800	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = 𝑥)
80	63, 79	jca 520	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))
81	80	ex 417	. . . . . 6 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
82	26, 81	sylbid 243	. . . . 5 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
83	82	com12 33	. . . 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 7853	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
86	6, 85	mpcom 39	1 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∀wral 3077  {crab 3414   ⊆ wss 3904  𝒫 cpw 4561  ∪ cuni 4871  ∩ cint 4911  dom cdm 5661   “ cima 5664  Ord word 6359  Oncon0 6360  Lim wlim 6361  suc csuc 6362  Fun wfun 6530  ‘cfv 6536  𝑅₁cr1 9733  rankcrnk 9734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-r1 9735  df-rank 9736

This theorem is referenced by:  rankonid  9800  onwf  9801  onssr1  9802