Theorem dfac12lem3 9282

Description: Lemma for dfac12 9286. (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 6446	. . . 4 ⊢ (𝐺‘𝐴) ∈ V
2	1	rnex 7362	. . 3 ⊢ ran (𝐺‘𝐴) ∈ V
3		ssid 3848	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4		dfac12.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On)
5		sseq1 3851	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 ⊆ 𝐴 ↔ 𝑛 ⊆ 𝐴))
6		fveq2 6433	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝐺‘𝑚) = (𝐺‘𝑛))
7		f1eq1 6333	. . . . . . . . . . 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 6433	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑅1‘𝑚) = (𝑅1‘𝑛))
10		f1eq2 6334	. . . . . . . . . . 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 271	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
13	5, 12	imbi12d 336	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On) ↔ (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On)))
14	13	imbi2d 332	. . . . . . 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 3851	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → (𝑚 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
16		fveq2 6433	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝐺‘𝑚) = (𝐺‘𝐴))
17		f1eq1 6333	. . . . . . . . . . 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 6433	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝑅1‘𝑚) = (𝑅1‘𝐴))
20		f1eq2 6334	. . . . . . . . . . 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 271	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
23	15, 22	imbi12d 336	. . . . . . . 8 ⊢ (𝑚 = 𝐴 → ((𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On) ↔ (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)))
24	23	imbi2d 332	. . . . . . 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 3169	. . . . . . . 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 5973	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ On → Ord 𝑚)
27	26	ad2antrl 721	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) → Ord 𝑚)
28		ordelss 5979	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝑚 ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
29	27, 28	sylan 577	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
30		simplrr 798	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑚 ⊆ 𝐴)
31	29, 30	sstrd 3837	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝐴)
32		pm5.5 353	. . . . . . . . . . . . . . 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 3194	. . . . . . . . . . . . 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 719	. . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . . 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 797	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝑚 ∈ On)
40		eqid 2825	. . . . . . . . . . . . . . 15 ⊢ (◡OrdIso( E , ran (𝐺‘∪ 𝑚)) ∘ (𝐺‘∪ 𝑚)) = (◡OrdIso( E , ran (𝐺‘∪ 𝑚)) ∘ (𝐺‘∪ 𝑚))
41		simplrr 798	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝑚 ⊆ 𝐴)
42		simpr 479	. . . . . . . . . . . . . . . 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 6433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
44		f1eq1 6333	. . . . . . . . . . . . . . . . . . 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 6433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝑅1‘𝑛) = (𝑅1‘𝑧))
47		f1eq2 6334	. . . . . . . . . . . . . . . . . . 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 271	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
50	49	cbvralv 3383	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ ∀𝑧 ∈ 𝑚 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
51	42, 50	sylib 210	. . . . . . . . . . . . . . 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 9281	. . . . . . . . . . . . . 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 403	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 450	. . . . . . . . . . 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 404	. . . . . . . . 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 234	. . . . . . 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 7318	. . . . . 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 6338	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On → (𝐺‘𝐴):(𝑅1‘𝐴)⟶On)
64		frn 6284	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)⟶On → ran (𝐺‘𝐴) ⊆ On)
65	62, 63, 64	3syl 18	. . 3 ⊢ (𝜑 → ran (𝐺‘𝐴) ⊆ On)
66		onssnum 9176	. . 3 ⊢ ((ran (𝐺‘𝐴) ∈ V ∧ ran (𝐺‘𝐴) ⊆ On) → ran (𝐺‘𝐴) ∈ dom card)
67	2, 65, 66	sylancr 583	. 2 ⊢ (𝜑 → ran (𝐺‘𝐴) ∈ dom card)
68		f1f1orn 6389	. . . 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 6446	. . . 4 ⊢ (𝑅1‘𝐴) ∈ V
71	70	f1oen 8243	. . 3 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴) → (𝑅1‘𝐴) ≈ ran (𝐺‘𝐴))
72		ennum 9086	. . 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 249	1 ⊢ (𝜑 → (𝑅1‘𝐴) ∈ dom card)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3117  Vcvv 3414   ⊆ wss 3798  ifcif 4306  𝒫 cpw 4378  ∪ cuni 4658   class class class wbr 4873   ↦ cmpt 4952   E cep 5254  ^◡ccnv 5341  dom cdm 5342  ran crn 5343   “ cima 5345   ∘ ccom 5346  Ord word 5962  Oncon0 5963  suc csuc 5965  ⟶wf 6119  –1-1→wf1 6120  –1-1-onto→wf1o 6122  ‘cfv 6123  (class class class)co 6905  recscrecs 7733   +_o coa 7823   ·_o comu 7824   ≈ cen 8219  OrdIsocoi 8683  harchar 8730  𝑅₁cr1 8902  rankcrnk 8903  cardccrd 9074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-oadd 7830  df-omul 7831  df-er 8009  df-en 8223  df-dom 8224  df-oi 8684  df-har 8732  df-r1 8904  df-rank 8905  df-card 9078

This theorem is referenced by:  dfac12r  9283