Theorem dfac12lem3 10129

Description: Lemma for dfac12 10133. (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 6895	. . . 4 ⊢ (𝐺‘𝐴) ∈ V
2	1	rnex 7907	. . 3 ⊢ ran (𝐺‘𝐴) ∈ V
3		ssid 3967	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4		dfac12.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On)
5		sseq1 3970	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 ⊆ 𝐴 ↔ 𝑛 ⊆ 𝐴))
6		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝐺‘𝑚) = (𝐺‘𝑛))
7		f1eq1 6770	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑚) = (𝐺‘𝑛) → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑚)–1-1→On))
8	6, 7	syl 18	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑚)–1-1→On))
9		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑅1‘𝑚) = (𝑅1‘𝑛))
10		f1eq2 6771	. . . . . . . . . . 11 ⊢ ((𝑅1‘𝑚) = (𝑅1‘𝑛) → ((𝐺‘𝑛):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
11	9, 10	syl 18	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝑛):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
12	8, 11	bitrd 282	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
13	5, 12	imbi12d 347	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On) ↔ (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On)))
14	13	imbi2d 343	. . . . . . 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 3970	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → (𝑚 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
16		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝐺‘𝑚) = (𝐺‘𝐴))
17		f1eq1 6770	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑚) = (𝐺‘𝐴) → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝑚)–1-1→On))
18	16, 17	syl 18	. . . . . . . . . 10 ⊢ (𝑚 = 𝐴 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝑚)–1-1→On))
19		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝑅1‘𝑚) = (𝑅1‘𝐴))
20		f1eq2 6771	. . . . . . . . . . 11 ⊢ ((𝑅1‘𝑚) = (𝑅1‘𝐴) → ((𝐺‘𝐴):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
21	19, 20	syl 18	. . . . . . . . . 10 ⊢ (𝑚 = 𝐴 → ((𝐺‘𝐴):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
22	18, 21	bitrd 282	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
23	15, 22	imbi12d 347	. . . . . . . 8 ⊢ (𝑚 = 𝐴 → ((𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On) ↔ (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)))
24	23	imbi2d 343	. . . . . . 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 3196	. . . . . . . 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 6371	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ On → Ord 𝑚)
27	26	ad2antrl 740	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) → Ord 𝑚)
28		ordelss 6377	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝑚 ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
29	27, 28	sylan 591	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
30		simplrr 789	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑚 ⊆ 𝐴)
31	29, 30	sstrd 3955	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝐴)
32		pm5.5 364	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ⊆ 𝐴 → ((𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
33	31, 32	syl 18	. . . . . . . . . . . . . 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 3192	. . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝑚 ∈ On)
40		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (◡OrdIso( E , ran (𝐺‘∪ 𝑚)) ∘ (𝐺‘∪ 𝑚)) = (◡OrdIso( E , ran (𝐺‘∪ 𝑚)) ∘ (𝐺‘∪ 𝑚))
41		simplrr 789	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝑚 ⊆ 𝐴)
42		fveq2 6882	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
43		f1eq1 6770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺‘𝑛) = (𝐺‘𝑧) → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On))
44	42, 43	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On))
45		fveq2 6882	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝑅1‘𝑛) = (𝑅1‘𝑧))
46		f1eq2 6771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅1‘𝑛) = (𝑅1‘𝑧) → ((𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
47	45, 46	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
48	44, 47	bitrd 282	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
49	48	cbvralvw 3249	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ ∀𝑧 ∈ 𝑚 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
50	49	bilani 509	. . . . . . . . . . . . . . 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 10128	. . . . . . . . . . . . . 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 417	. . . . . . . . . . . . 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 243	. . . . . . . . . . . 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 461	. . . . . . . . . . 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 87	. . . . . . . . . 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 418	. . . . . . . . 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 30	. . . . . . . 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 245	. . . . . . 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 7854	. . . . . 6 ⊢ (𝐴 ∈ On → (𝜑 → (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)))
60	4, 59	mpcom 39	. . . . 5 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
61	3, 60	mpi 21	. . . 4 ⊢ (𝜑 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)
62		f1f 6775	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On → (𝐺‘𝐴):(𝑅1‘𝐴)⟶On)
63		frn 6714	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)⟶On → ran (𝐺‘𝐴) ⊆ On)
64	61, 62, 63	3syl 19	. . 3 ⊢ (𝜑 → ran (𝐺‘𝐴) ⊆ On)
65		onssnum 10024	. . 3 ⊢ ((ran (𝐺‘𝐴) ∈ V ∧ ran (𝐺‘𝐴) ⊆ On) → ran (𝐺‘𝐴) ∈ dom card)
66	2, 64, 65	sylancr 598	. 2 ⊢ (𝜑 → ran (𝐺‘𝐴) ∈ dom card)
67		f1f1orn 6833	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴))
68	61, 67	syl 18	. . 3 ⊢ (𝜑 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴))
69		fvex 6895	. . . 4 ⊢ (𝑅1‘𝐴) ∈ V
70	69	f1oen 8969	. . 3 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴) → (𝑅1‘𝐴) ≈ ran (𝐺‘𝐴))
71		ennum 9933	. . 3 ⊢ ((𝑅1‘𝐴) ≈ ran (𝐺‘𝐴) → ((𝑅1‘𝐴) ∈ dom card ↔ ran (𝐺‘𝐴) ∈ dom card))
72	68, 70, 71	3syl 19	. 2 ⊢ (𝜑 → ((𝑅1‘𝐴) ∈ dom card ↔ ran (𝐺‘𝐴) ∈ dom card))
73	66, 72	mpbird 260	1 ⊢ (𝜑 → (𝑅1‘𝐴) ∈ dom card)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ifcif 4492  𝒫 cpw 4567  ∪ cuni 4876   class class class wbr 5113   ↦ cmpt 5196   E cep 5561  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   “ cima 5665   ∘ ccom 5666  Ord word 6360  Oncon0 6361  suc csuc 6363  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  recscrecs 8357   +_o coa 8450   ·_o comu 8451   ≈ cen 8940  OrdIsocoi 9471  harchar 9518  𝑅₁cr1 9734  rankcrnk 9735  cardccrd 9921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-oadd 8457  df-omul 8458  df-er 8694  df-en 8944  df-dom 8945  df-oi 9472  df-har 9519  df-r1 9736  df-rank 9737  df-card 9925

This theorem is referenced by:  dfac12r  10130