Theorem cff1 10208

Description: There is always a map from (cf‘𝐴) to 𝐴 (this is a stronger condition than the definition, which only presupposes a map from some 𝑦 ≈ (cf‘𝐴). (Contributed by Mario Carneiro, 28-Feb-2013.)

Assertion

Ref	Expression
cff1	⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))

Distinct variable group:   𝐴,𝑓,𝑤,𝑧

Proof of Theorem cff1

Dummy variables 𝑠 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfval 10196	. . . 4 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
2		cardon 9895	. . . . . . . . 9 ⊢ (card‘𝑦) ∈ On
3		eleq1 2849	. . . . . . . . 9 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
4	2, 3	mpbiri 260	. . . . . . . 8 ⊢ (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
5	4	adantr 484	. . . . . . 7 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑥 ∈ On)
6	5	exlimiv 1949	. . . . . 6 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑥 ∈ On)
7	6	abssi 4019	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ⊆ On
8		cflem 10194	. . . . . 6 ⊢ (𝐴 ∈ On → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
9		abn0 4335	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ≠ ∅ ↔ ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
10	8, 9	sylibr 236	. . . . 5 ⊢ (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ≠ ∅)
11		onint 7767	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ⊆ On ∧ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ≠ ∅) → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
12	7, 10, 11	sylancr 596	. . . 4 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
13	1, 12	eqeltrd 2861	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
14		fvex 6874	. . . 4 ⊢ (cf‘𝐴) ∈ V
15		eqeq1 2765	. . . . . 6 ⊢ (𝑥 = (cf‘𝐴) → (𝑥 = (card‘𝑦) ↔ (cf‘𝐴) = (card‘𝑦)))
16	15	anbi1d 640	. . . . 5 ⊢ (𝑥 = (cf‘𝐴) → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) ↔ ((cf‘𝐴) = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))))
17	16	exbidv 1940	. . . 4 ⊢ (𝑥 = (cf‘𝐴) → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))))
18	14, 17	elab 3637	. . 3 ⊢ ((cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
19	13, 18	sylib 220	. 2 ⊢ (𝐴 ∈ On → ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
20		simplr 778	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → (cf‘𝐴) = (card‘𝑦))
21		onss 7762	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
22		sstr 3942	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑦 ⊆ On)
23	21, 22	sylan2 602	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ∈ On) → 𝑦 ⊆ On)
24	23	ancoms 462	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
25	24	ad2ant2r 757	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑦 ⊆ On)
26		vex 3457	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
27		onssnum 9989	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
28	26, 27	mpan 700	. . . . . . . . . 10 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
29		cardid2 9904	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
30	28, 29	syl 17	. . . . . . . . 9 ⊢ (𝑦 ⊆ On → (card‘𝑦) ≈ 𝑦)
31	30	adantl 485	. . . . . . . 8 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (card‘𝑦) ≈ 𝑦)
32		breq1 5100	. . . . . . . . 9 ⊢ ((cf‘𝐴) = (card‘𝑦) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
33	32	adantr 484	. . . . . . . 8 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
34	31, 33	mpbird 259	. . . . . . 7 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (cf‘𝐴) ≈ 𝑦)
35		bren 8930	. . . . . . 7 ⊢ ((cf‘𝐴) ≈ 𝑦 ↔ ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto→𝑦)
36	34, 35	sylib 220	. . . . . 6 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto→𝑦)
37	20, 25, 36	syl2anc 593	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto→𝑦)
38		f1of1 6799	. . . . . . . . . . 11 ⊢ (𝑓:(cf‘𝐴)–1-1-onto→𝑦 → 𝑓:(cf‘𝐴)–1-1→𝑦)
39		f1ss 6761	. . . . . . . . . . . 12 ⊢ ((𝑓:(cf‘𝐴)–1-1→𝑦 ∧ 𝑦 ⊆ 𝐴) → 𝑓:(cf‘𝐴)–1-1→𝐴)
40	39	ancoms 462	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑓:(cf‘𝐴)–1-1→𝑦) → 𝑓:(cf‘𝐴)–1-1→𝐴)
41	38, 40	sylan2 602	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑓:(cf‘𝐴)–1-1-onto→𝑦) → 𝑓:(cf‘𝐴)–1-1→𝐴)
42	41	adantlr 725	. . . . . . . . 9 ⊢ (((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto→𝑦) → 𝑓:(cf‘𝐴)–1-1→𝐴)
43	42	3adant1 1142	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto→𝑦) → 𝑓:(cf‘𝐴)–1-1→𝐴)
44		f1ofo 6808	. . . . . . . . . . . 12 ⊢ (𝑓:(cf‘𝐴)–1-1-onto→𝑦 → 𝑓:(cf‘𝐴)–onto→𝑦)
45		foelrn 7082	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘𝐴)–onto→𝑦 ∧ 𝑠 ∈ 𝑦) → ∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓‘𝑤))
46		sseq2 3960	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑠 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
47	46	biimpcd 251	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ 𝑠 → (𝑠 = (𝑓‘𝑤) → 𝑧 ⊆ (𝑓‘𝑤)))
48	47	reximdv 3176	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ 𝑠 → (∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓‘𝑤) → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
49	45, 48	syl5com 31	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(cf‘𝐴)–onto→𝑦 ∧ 𝑠 ∈ 𝑦) → (𝑧 ⊆ 𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
50	49	rexlimdva 3162	. . . . . . . . . . . . 13 ⊢ (𝑓:(cf‘𝐴)–onto→𝑦 → (∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
51	50	ralimdv 3175	. . . . . . . . . . . 12 ⊢ (𝑓:(cf‘𝐴)–onto→𝑦 → (∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
52	44, 51	syl 17	. . . . . . . . . . 11 ⊢ (𝑓:(cf‘𝐴)–1-1-onto→𝑦 → (∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
53	52	impcom 411	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 ∧ 𝑓:(cf‘𝐴)–1-1-onto→𝑦) → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))
54	53	adantll 724	. . . . . . . . 9 ⊢ (((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto→𝑦) → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))
55	54	3adant1 1142	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto→𝑦) → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))
56	43, 55	jca 519	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto→𝑦) → (𝑓:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
57	56	3expia 1133	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → (𝑓:(cf‘𝐴)–1-1-onto→𝑦 → (𝑓:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))))
58	57	eximdv 1936	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → (∃𝑓 𝑓:(cf‘𝐴)–1-1-onto→𝑦 → ∃𝑓(𝑓:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))))
59	37, 58	mpd 15	. . . 4 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → ∃𝑓(𝑓:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
60	59	expl 461	. . 3 ⊢ (𝐴 ∈ On → (((cf‘𝐴) = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → ∃𝑓(𝑓:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))))
61	60	exlimdv 1952	. 2 ⊢ (𝐴 ∈ On → (∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → ∃𝑓(𝑓:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))))
62	19, 61	mpd 15	1 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ⊆ wss 3902  ∅c0 4283  ∩ cint 4902   class class class wbr 5097  dom cdm 5643  Oncon0 6340  –1-1→wf1 6512  –onto→wfo 6513  –1-1-onto→wf1o 6514  ‘cfv 6515   ≈ cen 8917  cardccrd 9886  cfccf 9888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-isom 6524  df-riota 7347  df-ov 7393  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-er 8671  df-en 8921  df-dom 8922  df-card 9890  df-cf 9892

This theorem is referenced by:  cfsmolem  10220  cfcoflem  10222  cfcof  10224  alephreg  10533