Theorem ackbij1lem18 8942

Description: Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Hypothesis

Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))

Assertion

Ref	Expression
ackbij1lem18	⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴))

Distinct variable groups:   𝐹,𝑏,𝑥,𝑦   𝐴,𝑏,𝑥,𝑦

Proof of Theorem ackbij1lem18

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difss 3699	. . . 4 ⊢ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴
2		ackbij.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
3	2	ackbij1lem11 8935	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴) → (𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
4	1, 3	mpan2 703	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
5		difss 3699	. . . . . . 7 ⊢ (ω ∖ 𝐴) ⊆ ω
6		omsson 6961	. . . . . . 7 ⊢ ω ⊆ On
7	5, 6	sstri 3577	. . . . . 6 ⊢ (ω ∖ 𝐴) ⊆ On
8		ominf 8057	. . . . . . . 8 ⊢ ¬ ω ∈ Fin
9		inss2 3796	. . . . . . . . 9 ⊢ (𝒫 ω ∩ Fin) ⊆ Fin
10	9	sseli 3564	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
11		difinf 8115	. . . . . . . 8 ⊢ ((¬ ω ∈ Fin ∧ 𝐴 ∈ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
12	8, 10, 11	sylancr 694	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
13		0fin 8073	. . . . . . . . 9 ⊢ ∅ ∈ Fin
14		eleq1 2676	. . . . . . . . 9 ⊢ ((ω ∖ 𝐴) = ∅ → ((ω ∖ 𝐴) ∈ Fin ↔ ∅ ∈ Fin))
15	13, 14	mpbiri 247	. . . . . . . 8 ⊢ ((ω ∖ 𝐴) = ∅ → (ω ∖ 𝐴) ∈ Fin)
16	15	necon3bi 2808	. . . . . . 7 ⊢ (¬ (ω ∖ 𝐴) ∈ Fin → (ω ∖ 𝐴) ≠ ∅)
17	12, 16	syl 17	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ≠ ∅)
18		onint 6887	. . . . . 6 ⊢ (((ω ∖ 𝐴) ⊆ On ∧ (ω ∖ 𝐴) ≠ ∅) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
19	7, 17, 18	sylancr 694	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
20	19	eldifad 3552	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ ω)
21		ackbij1lem4 8928	. . . 4 ⊢ (∩ (ω ∖ 𝐴) ∈ ω → {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
22	20, 21	syl 17	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
23		ackbij1lem6 8930	. . 3 ⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) ∧ {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin)) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin))
24	4, 22, 23	syl2anc 691	. 2 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin))
25	19	eldifbd 3553	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴)
26		disjsn 4192	. . . . . 6 ⊢ ((𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅ ↔ ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴)
27	25, 26	sylibr 223	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅)
28		ssdisj 3978	. . . . 5 ⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴 ∧ (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩ (ω ∖ 𝐴)}) = ∅)
29	1, 27, 28	sylancr 694	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩ (ω ∖ 𝐴)}) = ∅)
30	2	ackbij1lem9 8933	. . . 4 ⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) ∧ {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin) ∧ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩ (ω ∖ 𝐴)}) = ∅) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘{∩ (ω ∖ 𝐴)})))
31	4, 22, 29, 30	syl3anc 1318	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘{∩ (ω ∖ 𝐴)})))
32	2	ackbij1lem14 8938	. . . . 5 ⊢ (∩ (ω ∖ 𝐴) ∈ ω → (𝐹‘{∩ (ω ∖ 𝐴)}) = suc (𝐹‘∩ (ω ∖ 𝐴)))
33	20, 32	syl 17	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘{∩ (ω ∖ 𝐴)}) = suc (𝐹‘∩ (ω ∖ 𝐴)))
34	33	oveq2d 6565	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘{∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 suc (𝐹‘∩ (ω ∖ 𝐴))))
35	2	ackbij1lem10 8934	. . . . . . 7 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
36	35	ffvelrni 6266	. . . . . 6 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω)
37	4, 36	syl 17	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω)
38		ackbij1lem3 8927	. . . . . . 7 ⊢ (∩ (ω ∖ 𝐴) ∈ ω → ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin))
39	20, 38	syl 17	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin))
40	35	ffvelrni 6266	. . . . . 6 ⊢ (∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin) → (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω)
41	39, 40	syl 17	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω)
42		nnasuc 7573	. . . . 5 ⊢ (((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω ∧ (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 suc (𝐹‘∩ (ω ∖ 𝐴))) = suc ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘∩ (ω ∖ 𝐴))))
43	37, 41, 42	syl2anc 691	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 suc (𝐹‘∩ (ω ∖ 𝐴))) = suc ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘∩ (ω ∖ 𝐴))))
44		incom 3767	. . . . . . . . 9 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = (∩ (ω ∖ 𝐴) ∩ (𝐴 ∖ ∩ (ω ∖ 𝐴)))
45		disjdif 3992	. . . . . . . . 9 ⊢ (∩ (ω ∖ 𝐴) ∩ (𝐴 ∖ ∩ (ω ∖ 𝐴))) = ∅
46	44, 45	eqtri 2632	. . . . . . . 8 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = ∅
47	46	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = ∅)
48	2	ackbij1lem9 8933	. . . . . . 7 ⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) ∧ ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin) ∧ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩ (ω ∖ 𝐴)) = ∅) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘∩ (ω ∖ 𝐴))))
49	4, 39, 47, 48	syl3anc 1318	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘∩ (ω ∖ 𝐴))))
50		uncom 3719	. . . . . . . 8 ⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴)) = (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴)))
51		onnmin 6895	. . . . . . . . . . . . . . 15 ⊢ (((ω ∖ 𝐴) ⊆ On ∧ 𝑎 ∈ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴))
52	7, 51	mpan 702	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (ω ∖ 𝐴) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴))
53	52	con2i 133	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ∩ (ω ∖ 𝐴) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
54	53	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
55		ordom 6966	. . . . . . . . . . . . . . 15 ⊢ Ord ω
56		ordelss 5656	. . . . . . . . . . . . . . 15 ⊢ ((Ord ω ∧ ∩ (ω ∖ 𝐴) ∈ ω) → ∩ (ω ∖ 𝐴) ⊆ ω)
57	55, 20, 56	sylancr 694	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ⊆ ω)
58	57	sselda 3568	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ ω)
59		eldif 3550	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (ω ∖ 𝐴) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ 𝐴))
60	59	simplbi2 653	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ω → (¬ 𝑎 ∈ 𝐴 → 𝑎 ∈ (ω ∖ 𝐴)))
61	60	orrd 392	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ω → (𝑎 ∈ 𝐴 ∨ 𝑎 ∈ (ω ∖ 𝐴)))
62	61	orcomd 402	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ω → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴))
63	58, 62	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴))
64		orel1 396	. . . . . . . . . . . 12 ⊢ (¬ 𝑎 ∈ (ω ∖ 𝐴) → ((𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴))
65	54, 63, 64	sylc 63	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ 𝐴)
66	65	ex 449	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝑎 ∈ ∩ (ω ∖ 𝐴) → 𝑎 ∈ 𝐴))
67	66	ssrdv 3574	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∩ (ω ∖ 𝐴) ⊆ 𝐴)
68		undif 4001	. . . . . . . . 9 ⊢ (∩ (ω ∖ 𝐴) ⊆ 𝐴 ↔ (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴)
69	67, 68	sylib 207	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴)
70	50, 69	syl5eq 2656	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴)) = 𝐴)
71	70	fveq2d 6107	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩ (ω ∖ 𝐴))) = (𝐹‘𝐴))
72	49, 71	eqtr3d 2646	. . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘∩ (ω ∖ 𝐴))) = (𝐹‘𝐴))
73		suceq 5707	. . . . 5 ⊢ (((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘∩ (ω ∖ 𝐴))) = (𝐹‘𝐴) → suc ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘∩ (ω ∖ 𝐴))) = suc (𝐹‘𝐴))
74	72, 73	syl 17	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → suc ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 (𝐹‘∩ (ω ∖ 𝐴))) = suc (𝐹‘𝐴))
75	43, 74	eqtrd 2644	. . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 suc (𝐹‘∩ (ω ∖ 𝐴))) = suc (𝐹‘𝐴))
76	31, 34, 75	3eqtrd 2648	. 2 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴))
77		fveq2 6103	. . . 4 ⊢ (𝑏 = ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) → (𝐹‘𝑏) = (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})))
78	77	eqeq1d 2612	. . 3 ⊢ (𝑏 = ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) → ((𝐹‘𝑏) = suc (𝐹‘𝐴) ↔ (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴)))
79	78	rspcev 3282	. 2 ⊢ ((((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin) ∧ (𝐹‘((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴)) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴))
80	24, 76, 79	syl2anc 691	1 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∩ cint 4410  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  Ord word 5639  Oncon0 5640  suc csuc 5642  ‘cfv 5804  (class class class)co 6549  ωcom 6957   +_𝑜 coa 7444  Fincfn 7841  cardccrd 8644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873

This theorem is referenced by:  ackbij1  8943