Theorem dfac12lem3 9638

Description: Lemma for dfac12 9642. (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 6681	. . . 4 ⊢ (𝐺‘𝐴) ∈ V
2	1	rnex 7636	. . 3 ⊢ ran (𝐺‘𝐴) ∈ V
3		ssid 3897	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4		dfac12.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On)
5		sseq1 3900	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 ⊆ 𝐴 ↔ 𝑛 ⊆ 𝐴))
6		fveq2 6668	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝐺‘𝑚) = (𝐺‘𝑛))
7		f1eq1 6563	. . . . . . . . . . 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 6668	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑅1‘𝑚) = (𝑅1‘𝑛))
10		f1eq2 6564	. . . . . . . . . . 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 282	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On))
13	5, 12	imbi12d 348	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On) ↔ (𝑛 ⊆ 𝐴 → (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On)))
14	13	imbi2d 344	. . . . . . 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 3900	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → (𝑚 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
16		fveq2 6668	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝐺‘𝑚) = (𝐺‘𝐴))
17		f1eq1 6563	. . . . . . . . . . 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 6668	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝑅1‘𝑚) = (𝑅1‘𝐴))
20		f1eq2 6564	. . . . . . . . . . 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 282	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → ((𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On ↔ (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On))
23	15, 22	imbi12d 348	. . . . . . . 8 ⊢ (𝑚 = 𝐴 → ((𝑚 ⊆ 𝐴 → (𝐺‘𝑚):(𝑅1‘𝑚)–1-1→On) ↔ (𝐴 ⊆ 𝐴 → (𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On)))
24	23	imbi2d 344	. . . . . . 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 3089	. . . . . . . 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 6176	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ On → Ord 𝑚)
27	26	ad2antrl 728	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) → Ord 𝑚)
28		ordelss 6182	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝑚 ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
29	27, 28	sylan 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
30		simplrr 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑚 ⊆ 𝐴)
31	29, 30	sstrd 3885	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝐴)
32		pm5.5 365	. . . . . . . . . . . . . . 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 3108	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On) → 𝑚 ⊆ 𝐴)
42		simpr 488	. . . . . . . . . . . . . . . 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 6668	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
44		f1eq1 6563	. . . . . . . . . . . . . . . . . . 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 6668	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝑅1‘𝑛) = (𝑅1‘𝑧))
47		f1eq2 6564	. . . . . . . . . . . . . . . . . . 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 282	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On))
50	49	cbvralvw 3348	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅1‘𝑛)–1-1→On ↔ ∀𝑧 ∈ 𝑚 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
51	42, 50	sylib 221	. . . . . . . . . . . . . . 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 9637	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . . 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 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))
55	54	expr 460	. . . . . . . . . . 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 417	. . . . . . . . 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 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))))
60	14, 24, 59	tfis3 7585	. . . . . 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 6568	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1→On → (𝐺‘𝐴):(𝑅1‘𝐴)⟶On)
64		frn 6505	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)⟶On → ran (𝐺‘𝐴) ⊆ On)
65	62, 63, 64	3syl 18	. . 3 ⊢ (𝜑 → ran (𝐺‘𝐴) ⊆ On)
66		onssnum 9533	. . 3 ⊢ ((ran (𝐺‘𝐴) ∈ V ∧ ran (𝐺‘𝐴) ⊆ On) → ran (𝐺‘𝐴) ∈ dom card)
67	2, 65, 66	sylancr 590	. 2 ⊢ (𝜑 → ran (𝐺‘𝐴) ∈ dom card)
68		f1f1orn 6623	. . . 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 6681	. . . 4 ⊢ (𝑅1‘𝐴) ∈ V
71	70	f1oen 8569	. . 3 ⊢ ((𝐺‘𝐴):(𝑅1‘𝐴)–1-1-onto→ran (𝐺‘𝐴) → (𝑅1‘𝐴) ≈ ran (𝐺‘𝐴))
72		ennum 9442	. . 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 260	1 ⊢ (𝜑 → (𝑅1‘𝐴) ∈ dom card)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2113  ∀wral 3053  Vcvv 3397   ⊆ wss 3841  ifcif 4411  𝒫 cpw 4485  ∪ cuni 4793   class class class wbr 5027   ↦ cmpt 5107   E cep 5429  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522   ∘ ccom 5523  Ord word 6165  Oncon0 6166  suc csuc 6168  ⟶wf 6329  –1-1→wf1 6330  –1-1-onto→wf1o 6332  ‘cfv 6333  (class class class)co 7164  recscrecs 8029   +_o coa 8121   ·_o comu 8122   ≈ cen 8545  OrdIsocoi 9039  harchar 9086  𝑅₁cr1 9257  rankcrnk 9258  cardccrd 9430

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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-isom 6342  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-oadd 8128  df-omul 8129  df-er 8313  df-en 8549  df-dom 8550  df-oi 9040  df-har 9087  df-r1 9259  df-rank 9260  df-card 9434

This theorem is referenced by:  dfac12r  9639