Theorem cff1 10217

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 10206	. . . 4 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
2		cardon 9903	. . . . . . . . 9 ⊢ (card‘𝑦) ∈ On
3		eleq1 2817	. . . . . . . . 9 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
4	2, 3	mpbiri 258	. . . . . . . 8 ⊢ (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑥 ∈ On)
6	5	exlimiv 1930	. . . . . 6 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑥 ∈ On)
7	6	abssi 4035	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ⊆ On
8		cflem 10204	. . . . . 6 ⊢ (𝐴 ∈ On → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
9		abn0 4350	. . . . . 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 234	. . . . 5 ⊢ (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ≠ ∅)
11		onint 7768	. . . . 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 587	. . . 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 2829	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
14		fvex 6873	. . . 4 ⊢ (cf‘𝐴) ∈ V
15		eqeq1 2734	. . . . . 6 ⊢ (𝑥 = (cf‘𝐴) → (𝑥 = (card‘𝑦) ↔ (cf‘𝐴) = (card‘𝑦)))
16	15	anbi1d 631	. . . . 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 1921	. . . 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 3648	. . 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 218	. 2 ⊢ (𝐴 ∈ On → ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
20		simplr 768	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → (cf‘𝐴) = (card‘𝑦))
21		onss 7763	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
22		sstr 3957	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑦 ⊆ On)
23	21, 22	sylan2 593	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ∈ On) → 𝑦 ⊆ On)
24	23	ancoms 458	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
25	24	ad2ant2r 747	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑦 ⊆ On)
26		vex 3454	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
27		onssnum 9999	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
28	26, 27	mpan 690	. . . . . . . . . 10 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
29		cardid2 9912	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
30	28, 29	syl 17	. . . . . . . . 9 ⊢ (𝑦 ⊆ On → (card‘𝑦) ≈ 𝑦)
31	30	adantl 481	. . . . . . . 8 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (card‘𝑦) ≈ 𝑦)
32		breq1 5112	. . . . . . . . 9 ⊢ ((cf‘𝐴) = (card‘𝑦) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
33	32	adantr 480	. . . . . . . 8 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
34	31, 33	mpbird 257	. . . . . . 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 218	. . . . . 6 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto→𝑦)
37	20, 25, 36	syl2anc 584	. . . . 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 6801	. . . . . . . . . . 11 ⊢ (𝑓:(cf‘𝐴)–1-1-onto→𝑦 → 𝑓:(cf‘𝐴)–1-1→𝑦)
39		f1ss 6763	. . . . . . . . . . . 12 ⊢ ((𝑓:(cf‘𝐴)–1-1→𝑦 ∧ 𝑦 ⊆ 𝐴) → 𝑓:(cf‘𝐴)–1-1→𝐴)
40	39	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑓:(cf‘𝐴)–1-1→𝑦) → 𝑓:(cf‘𝐴)–1-1→𝐴)
41	38, 40	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑓:(cf‘𝐴)–1-1-onto→𝑦) → 𝑓:(cf‘𝐴)–1-1→𝐴)
42	41	adantlr 715	. . . . . . . . 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 1130	. . . . . . . 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 6809	. . . . . . . . . . . 12 ⊢ (𝑓:(cf‘𝐴)–1-1-onto→𝑦 → 𝑓:(cf‘𝐴)–onto→𝑦)
45		foelrn 7081	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘𝐴)–onto→𝑦 ∧ 𝑠 ∈ 𝑦) → ∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓‘𝑤))
46		sseq2 3975	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑠 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
47	46	biimpcd 249	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ 𝑠 → (𝑠 = (𝑓‘𝑤) → 𝑧 ⊆ (𝑓‘𝑤)))
48	47	reximdv 3149	. . . . . . . . . . . . . . 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 3135	. . . . . . . . . . . . 13 ⊢ (𝑓:(cf‘𝐴)–onto→𝑦 → (∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
51	50	ralimdv 3148	. . . . . . . . . . . 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 407	. . . . . . . . . 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 714	. . . . . . . . 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 1130	. . . . . . . 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 511	. . . . . . 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 1121	. . . . . 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 1917	. . . . 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 457	. . 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 1933	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ⊆ wss 3916  ∅c0 4298  ∩ cint 4912   class class class wbr 5109  dom cdm 5640  Oncon0 6334  –1-1→wf1 6510  –onto→wfo 6511  –1-1-onto→wf1o 6512  ‘cfv 6513   ≈ cen 8917  cardccrd 9894  cfccf 9896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-isom 6522  df-riota 7346  df-ov 7392  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-er 8673  df-en 8921  df-dom 8922  df-card 9898  df-cf 9900

This theorem is referenced by:  cfsmolem  10229  cfcoflem  10231  cfcof  10233  alephreg  10541