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Theorem dfac12lem3 10140

Description: Lemma for dfac12 10144. (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 6905	. . . 4 ⊢ (𝐺‘𝐴) ∈ V
2	1	rnex 7903	. . 3 ⊢ ran (𝐺‘𝐴) ∈ V
3		ssid 4005	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4		dfac12.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On)
5		sseq1 4008	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 ⊆ 𝐴 ↔ 𝑛 ⊆ 𝐴))
6		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝐺‘𝑚) = (𝐺‘𝑛))
7		f1eq1 6783	. . . . . . . . . . 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 6892	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑅₁‘𝑚) = (𝑅₁‘𝑛))
10		f1eq2 6784	. . . . . . . . . . 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 279	. . . . . . . . 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 4008	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → (𝑚 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
16		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝐺‘𝑚) = (𝐺‘𝐴))
17		f1eq1 6783	. . . . . . . . . . 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 6892	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐴 → (𝑅₁‘𝑚) = (𝑅₁‘𝐴))
20		f1eq2 6784	. . . . . . . . . . 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 279	. . . . . . . . 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 3180	. . . . . . . 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 6375	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ On → Ord 𝑚)
27	26	ad2antrl 727	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) → Ord 𝑚)
28		ordelss 6381	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝑚 ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
29	27, 28	sylan 581	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑛 ⊆ 𝑚)
30		simplrr 777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ 𝑛 ∈ 𝑚) → 𝑚 ⊆ 𝐴)
31	29, 30	sstrd 3993	. . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅₁‘𝑛)–1-1→On) → 𝑚 ∈ On)
40		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (^◡OrdIso( E , ran (𝐺‘∪ 𝑚)) ∘ (𝐺‘∪ 𝑚)) = (^◡OrdIso( E , ran (𝐺‘∪ 𝑚)) ∘ (𝐺‘∪ 𝑚))
41		simplrr 777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚 ⊆ 𝐴)) ∧ ∀𝑛 ∈ 𝑚 (𝐺‘𝑛):(𝑅₁‘𝑛)–1-1→On) → 𝑚 ⊆ 𝐴)
42		simpr 486	. . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
44		f1eq1 6783	. . . . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝑅₁‘𝑛) = (𝑅₁‘𝑧))
47		f1eq2 6784	. . . . . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝑛):(𝑅₁‘𝑛)–1-1→On ↔ (𝐺‘𝑧):(𝑅₁‘𝑧)–1-1→On))
50	49	cbvralvw 3235	. . . . . . . . . . . . . . . 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 10139	. . . . . . . . . . . . . 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 414	. . . . . . . . . . . . 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 458	. . . . . . . . . . 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 415	. . . . . . . . 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	biimtrid 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 7847	. . . . . 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 6788	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅₁‘𝐴)–1-1→On → (𝐺‘𝐴):(𝑅₁‘𝐴)⟶On)
64		frn 6725	. . . 4 ⊢ ((𝐺‘𝐴):(𝑅₁‘𝐴)⟶On → ran (𝐺‘𝐴) ⊆ On)
65	62, 63, 64	3syl 18	. . 3 ⊢ (𝜑 → ran (𝐺‘𝐴) ⊆ On)
66		onssnum 10035	. . 3 ⊢ ((ran (𝐺‘𝐴) ∈ V ∧ ran (𝐺‘𝐴) ⊆ On) → ran (𝐺‘𝐴) ∈ dom card)
67	2, 65, 66	sylancr 588	. 2 ⊢ (𝜑 → ran (𝐺‘𝐴) ∈ dom card)
68		f1f1orn 6845	. . . 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 6905	. . . 4 ⊢ (𝑅₁‘𝐴) ∈ V
71	70	f1oen 8969	. . 3 ⊢ ((𝐺‘𝐴):(𝑅₁‘𝐴)–1-1-onto→ran (𝐺‘𝐴) → (𝑅₁‘𝐴) ≈ ran (𝐺‘𝐴))
72		ennum 9942	. . 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 257	1 ⊢ (𝜑 → (𝑅₁‘𝐴) ∈ dom card)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ⊆ wss 3949 ifcif 4529 𝒫 cpw 4603 ∪ cuni 4909 class class class wbr 5149 ↦ cmpt 5232 E cep 5580 ^◡ccnv 5676 dom cdm 5677 ran crn 5678 “ cima 5680 ∘ ccom 5681 Ord word 6364 Oncon0 6365 suc csuc 6367 ⟶wf 6540 –1-1→wf1 6541 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 recscrecs 8370 +_o coa 8463 ·_o comu 8464 ≈ cen 8936 OrdIsocoi 9504 harchar 9551 𝑅₁cr1 9757 rankcrnk 9758 cardccrd 9930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-oadd 8470 df-omul 8471 df-er 8703 df-en 8940 df-dom 8941 df-oi 9505 df-har 9552 df-r1 9759 df-rank 9760 df-card 9934

This theorem is referenced by: dfac12r 10141

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