Theorem dfac12lem3 10059

Description: Lemma for dfac12 10063. (Contributed by Mario Carneiro, 29-May-2015.)

Hypotheses

Ref	Expression
dfac12.1	⊢ (𝜑 → 𝐴 ∈ On)
dfac12.3	⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On)
dfac12.4	⊢ 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))))))

Assertion

Ref	Expression
dfac12lem3	⊢ (𝜑 → (𝑅1‘𝐴) ∈ dom card)

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐺   𝜑,𝑦   𝑥,𝐹,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem dfac12lem3

Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6840	. . . 4 ⊢ (𝐺‘𝐴) ∈ V
2	1	rnex 7850	. . 3 ⊢ ran (𝐺‘𝐴) ∈ V
3		ssid 3937	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4		dfac12.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On)
5		sseq1 3940	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 ⊆ 𝐴 ↔ 𝑛 ⊆ 𝐴))
6		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝐺‘𝑚) = (𝐺‘𝑛))
7		f1eq1 6718	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑚) = (𝐺‘𝑛) → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑚)–1-1→On))
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑚)–1-1→On))
9		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑅1‘𝑚) = (𝑅1‘𝑛))
10		f1eq2 6719	. . . . . . . . . . 11 ⊢ ((𝑅1‘𝑚) = (𝑅1‘𝑛) → ((𝐺‘𝑛):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝑛):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
12	8, 11	bitrd 280	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
13	5, 12	imbi12d 345	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On) ↔ (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On)))
14	13	imbi2d 341	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝜑 → (𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On)) ↔ (𝜑 → (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))))
15		sseq1 3940	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → (𝑚 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
16		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝐺‘𝑚) = (𝐺‘𝐴))
17		f1eq1 6718	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑚) = (𝐺‘𝐴) → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝑚)–1-1→On))
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝑚 = 𝐴 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝑚)–1-1→On))
19		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝑅1‘𝑚) = (𝑅1‘𝐴))
20		f1eq2 6719	. . . . . . . . . . 11 ⊢ ((𝑅1‘𝑚) = (𝑅1‘𝐴) → ((𝐺‘𝐴):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝑚 = 𝐴 → ((𝐺‘𝐴):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
22	18, 21	bitrd 280	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
23	15, 22	imbi12d 345	. . . . . . . 8 ⊢ (𝑚 = 𝐴 → ((𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On) ↔ (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)))
24	23	imbi2d 341	. . . . . . 7 ⊢ (𝑚 = 𝐴 → ((𝜑 → (𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On)) ↔ (𝜑 → (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))))
25		r19.21v 3164	. . . . . . . 8 ⊢ (∀𝑛 ∈ 𝑚 (𝜑 → (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On)) ↔ (𝜑 → ∀𝑛 ∈ 𝑚 (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On)))
26		eloni 6320	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ On → Ord 𝑚)
27	26	ad2antrl 734	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) → Ord 𝑚)
28		ordelss 6326	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝑚 ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
29	27, 28	sylan 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
30		simplrr 783	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑚 ⊆ 𝐴)
31	29, 30	sstrd 3925	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝐴)
32		pm5.5 362	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ⊆ 𝐴 → ((𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
33	31, 32	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → ((𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
34	33	ralbidva 3160	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) → (∀𝑛 ∈ 𝑚 (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) ↔ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
35	4	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝐴 ∈ On)
36		dfac12.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On)
37	36	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On)
38		dfac12.4	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))))))
39		simplrl 782	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝑚 ∈ On)
40		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (◡OrdIso( E , ran (𝐺‘∪ 𝑚)) ∘ (𝐺‘∪ 𝑚)) = (◡OrdIso( E , ran (𝐺‘∪ 𝑚)) ∘ (𝐺‘∪ 𝑚))
41		simplrr 783	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝑚 ⊆ 𝐴)
42		fveq2 6827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
43		f1eq1 6718	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺‘𝑛) = (𝐺‘𝑧) → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On))
44	42, 43	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On))
45		fveq2 6827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝑅1‘𝑛) = (𝑅1‘𝑧))
46		f1eq2 6719	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅1‘𝑛) = (𝑅1‘𝑧) → ((𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
47	45, 46	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
48	44, 47	bitrd 280	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
49	48	cbvralvw 3217	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ ∀𝑧 ∈ 𝑚 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
50	49	bilani 505	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → ∀𝑧 ∈ 𝑚 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
51	35, 37, 38, 39, 40, 41, 50	dfac12lem2 10058	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On)
52	51	ex 413	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) → (∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On))
53	34, 52	sylbid 241	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) → (∀𝑛 ∈ 𝑚 (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On))
54	53	expr 457	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ On) → (𝑚 ⊆ 𝐴 → (∀𝑛 ∈ 𝑚 (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On)))
55	54	com23 86	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ On) → (∀𝑛 ∈ 𝑚 (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → (𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On)))
56	55	expcom 414	. . . . . . . . 9 ⊢ (𝑚 ∈ On → (𝜑 → (∀𝑛 ∈ 𝑚 (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → (𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On))))
57	56	a2d 29	. . . . . . . 8 ⊢ (𝑚 ∈ On → ((𝜑 → ∀𝑛 ∈ 𝑚 (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On)) → (𝜑 → (𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On))))
58	25, 57	biimtrid 243	. . . . . . 7 ⊢ (𝑚 ∈ On → (∀𝑛 ∈ 𝑚 (𝜑 → (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On)) → (𝜑 → (𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On))))
59	14, 24, 58	tfis3 7798	. . . . . 6 ⊢ (𝐴 ∈ On → (𝜑 → (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)))
60	4, 59	mpcom 38	. . . . 5 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
61	3, 60	mpi 20	. . . 4 ⊢ (𝜑 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)
62		f1f 6723	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On → (𝐺‘𝐴):(𝑅1‘𝐴)⟶On)
63		frn 6662	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)⟶On → ran (𝐺‘𝐴) ⊆ On)
64	61, 62, 63	3syl 18	. . 3 ⊢ (𝜑 → ran (𝐺‘𝐴) ⊆ On)
65		onssnum 9953	. . 3 ⊢ ((ran (𝐺‘𝐴) ∈ V ∧ ran (𝐺‘𝐴) ⊆ On) → ran (𝐺‘𝐴) ∈ dom card)
66	2, 64, 65	sylancr 593	. 2 ⊢ (𝜑 → ran (𝐺‘𝐴) ∈ dom card)
67		f1f1orn 6778	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴))
68	61, 67	syl 17	. . 3 ⊢ (𝜑 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴))
69		fvex 6840	. . . 4 ⊢ (𝑅1‘𝐴) ∈ V
70	69	f1oen 8909	. . 3 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴) → (𝑅1‘𝐴) ≈ ran (𝐺‘𝐴))
71		ennum 9862	. . 3 ⊢ ((𝑅1‘𝐴) ≈ ran (𝐺‘𝐴) → ((𝑅1‘𝐴) ∈ dom card ↔ ran (𝐺‘𝐴) ∈ dom card))
72	68, 70, 71	3syl 18	. 2 ⊢ (𝜑 → ((𝑅1‘𝐴) ∈ dom card ↔ ran (𝐺‘𝐴) ∈ dom card))
73	66, 72	mpbird 258	1 ⊢ (𝜑 → (𝑅1‘𝐴) ∈ dom card)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ⊆ wss 3883  ifcif 4454  𝒫 cpw 4529  ∪ cuni 4838   class class class wbr 5072   ↦ cmpt 5153   E cep 5517  ^◡ccnv 5617  dom cdm 5618  ran crn 5619   “ cima 5621   ∘ ccom 5622  Ord word 6309  Oncon0 6310  suc csuc 6312  ⟶wf 6481  –1-1→wf1 6482  –1-1-onto→wf1o 6484  ‘cfv 6485  (class class class)co 7356  recscrecs 8300   +_o coa 8392   ·_o comu 8393   ≈ cen 8880  OrdIsocoi 9414  harchar 9461  𝑅₁cr1 9677  rankcrnk 9678  cardccrd 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-oadd 8399  df-omul 8400  df-er 8633  df-en 8884  df-dom 8885  df-oi 9415  df-har 9462  df-r1 9679  df-rank 9680  df-card 9854

This theorem is referenced by:  dfac12r  10060