Theorem dfac12lem3 9832

Description: Lemma for dfac12 9836. (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 6769	. . . 4 ⊢ (𝐺‘𝐴) ∈ V
2	1	rnex 7733	. . 3 ⊢ ran (𝐺‘𝐴) ∈ V
3		ssid 3939	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4		dfac12.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On)
5		sseq1 3942	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 ⊆ 𝐴 ↔ 𝑛 ⊆ 𝐴))
6		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝐺‘𝑚) = (𝐺‘𝑛))
7		f1eq1 6649	. . . . . . . . . . 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 6756	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑅1‘𝑚) = (𝑅1‘𝑛))
10		f1eq2 6650	. . . . . . . . . . 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 278	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
13	5, 12	imbi12d 344	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On) ↔ (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On)))
14	13	imbi2d 340	. . . . . . 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 3942	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → (𝑚 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
16		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝐺‘𝑚) = (𝐺‘𝐴))
17		f1eq1 6649	. . . . . . . . . . 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 6756	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝑅1‘𝑚) = (𝑅1‘𝐴))
20		f1eq2 6650	. . . . . . . . . . 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 278	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
23	15, 22	imbi12d 344	. . . . . . . 8 ⊢ (𝑚 = 𝐴 → ((𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On) ↔ (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)))
24	23	imbi2d 340	. . . . . . 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 3100	. . . . . . . 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 6261	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ On → Ord 𝑚)
27	26	ad2antrl 724	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) → Ord 𝑚)
28		ordelss 6267	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝑚 ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
29	27, 28	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
30		simplrr 774	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑚 ⊆ 𝐴)
31	29, 30	sstrd 3927	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝐴)
32		pm5.5 361	. . . . . . . . . . . . . . 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 3119	. . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝑚 ∈ On)
40		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (◡OrdIso( E , ran (𝐺‘∪ 𝑚)) ∘ (𝐺‘∪ 𝑚)) = (◡OrdIso( E , ran (𝐺‘∪ 𝑚)) ∘ (𝐺‘∪ 𝑚))
41		simplrr 774	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝑚 ⊆ 𝐴)
42		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On)
43		fveq2 6756	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
44		f1eq1 6649	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺‘𝑛) = (𝐺‘𝑧) → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On))
45	43, 44	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On))
46		fveq2 6756	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝑅1‘𝑛) = (𝑅1‘𝑧))
47		f1eq2 6650	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅1‘𝑛) = (𝑅1‘𝑧) → ((𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
48	46, 47	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑧):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
49	45, 48	bitrd 278	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
50	49	cbvralvw 3372	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ ∀𝑧 ∈ 𝑚 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
51	42, 50	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → ∀𝑧 ∈ 𝑚 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
52	35, 37, 38, 39, 40, 41, 51	dfac12lem2 9831	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On)
53	52	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) → (∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On))
54	34, 53	sylbid 239	. . . . . . . . . . . 12 ⊢ ((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	expr 456	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ On) → (𝑚 ⊆ 𝐴 → (∀𝑛 ∈ 𝑚 (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On)))
56	55	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)))
57	56	expcom 413	. . . . . . . . 9 ⊢ (𝑚 ∈ On → (𝜑 → (∀𝑛 ∈ 𝑚 (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → (𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On))))
58	57	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))))
59	25, 58	syl5bi 241	. . . . . . 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))))
60	14, 24, 59	tfis3 7679	. . . . . 6 ⊢ (𝐴 ∈ On → (𝜑 → (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)))
61	4, 60	mpcom 38	. . . . 5 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
62	3, 61	mpi 20	. . . 4 ⊢ (𝜑 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)
63		f1f 6654	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On → (𝐺‘𝐴):(𝑅1‘𝐴)⟶On)
64		frn 6591	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)⟶On → ran (𝐺‘𝐴) ⊆ On)
65	62, 63, 64	3syl 18	. . 3 ⊢ (𝜑 → ran (𝐺‘𝐴) ⊆ On)
66		onssnum 9727	. . 3 ⊢ ((ran (𝐺‘𝐴) ∈ V ∧ ran (𝐺‘𝐴) ⊆ On) → ran (𝐺‘𝐴) ∈ dom card)
67	2, 65, 66	sylancr 586	. 2 ⊢ (𝜑 → ran (𝐺‘𝐴) ∈ dom card)
68		f1f1orn 6711	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴))
69	62, 68	syl 17	. . 3 ⊢ (𝜑 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴))
70		fvex 6769	. . . 4 ⊢ (𝑅1‘𝐴) ∈ V
71	70	f1oen 8716	. . 3 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴) → (𝑅1‘𝐴) ≈ ran (𝐺‘𝐴))
72		ennum 9636	. . 3 ⊢ ((𝑅1‘𝐴) ≈ ran (𝐺‘𝐴) → ((𝑅1‘𝐴) ∈ dom card ↔ ran (𝐺‘𝐴) ∈ dom card))
73	69, 71, 72	3syl 18	. 2 ⊢ (𝜑 → ((𝑅1‘𝐴) ∈ dom card ↔ ran (𝐺‘𝐴) ∈ dom card))
74	67, 73	mpbird 256	1 ⊢ (𝜑 → (𝑅1‘𝐴) ∈ dom card)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883  ifcif 4456  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070   ↦ cmpt 5153   E cep 5485  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583   ∘ ccom 5584  Ord word 6250  Oncon0 6251  suc csuc 6253  ⟶wf 6414  –1-1→wf1 6415  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  recscrecs 8172   +_o coa 8264   ·_o comu 8265   ≈ cen 8688  OrdIsocoi 9198  harchar 9245  𝑅₁cr1 9451  rankcrnk 9452  cardccrd 9624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-oadd 8271  df-omul 8272  df-er 8456  df-en 8692  df-dom 8693  df-oi 9199  df-har 9246  df-r1 9453  df-rank 9454  df-card 9628

This theorem is referenced by:  dfac12r  9833