Theorem cff1 9080

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 9069	. . . 4 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
2		cardon 8770	. . . . . . . . 9 ⊢ (card‘𝑦) ∈ On
3		eleq1 2689	. . . . . . . . 9 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
4	2, 3	mpbiri 248	. . . . . . . 8 ⊢ (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
5	4	adantr 481	. . . . . . 7 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑥 ∈ On)
6	5	exlimiv 1858	. . . . . 6 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑥 ∈ On)
7	6	abssi 3677	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ⊆ On
8		cflem 9068	. . . . . 6 ⊢ (𝐴 ∈ On → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
9		abn0 3954	. . . . . 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 224	. . . . 5 ⊢ (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ≠ ∅)
11		onint 6995	. . . . 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 695	. . . 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 2701	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
14		fvex 6201	. . . 4 ⊢ (cf‘𝐴) ∈ V
15		eqeq1 2626	. . . . . 6 ⊢ (𝑥 = (cf‘𝐴) → (𝑥 = (card‘𝑦) ↔ (cf‘𝐴) = (card‘𝑦)))
16	15	anbi1d 741	. . . . 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 1850	. . . 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 3350	. . 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 208	. 2 ⊢ (𝐴 ∈ On → ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
20		simplr 792	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → (cf‘𝐴) = (card‘𝑦))
21		onss 6990	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
22		sstr 3611	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑦 ⊆ On)
23	21, 22	sylan2 491	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ∈ On) → 𝑦 ⊆ On)
24	23	ancoms 469	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
25	24	ad2ant2r 783	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑦 ⊆ On)
26		vex 3203	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
27		onssnum 8863	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
28	26, 27	mpan 706	. . . . . . . . . 10 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
29		cardid2 8779	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
30	28, 29	syl 17	. . . . . . . . 9 ⊢ (𝑦 ⊆ On → (card‘𝑦) ≈ 𝑦)
31	30	adantl 482	. . . . . . . 8 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (card‘𝑦) ≈ 𝑦)
32		breq1 4656	. . . . . . . . 9 ⊢ ((cf‘𝐴) = (card‘𝑦) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
33	32	adantr 481	. . . . . . . 8 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
34	31, 33	mpbird 247	. . . . . . 7 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (cf‘𝐴) ≈ 𝑦)
35		bren 7964	. . . . . . 7 ⊢ ((cf‘𝐴) ≈ 𝑦 ↔ ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto→𝑦)
36	34, 35	sylib 208	. . . . . 6 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto→𝑦)
37	20, 25, 36	syl2anc 693	. . . . 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 6136	. . . . . . . . . . 11 ⊢ (𝑓:(cf‘𝐴)–1-1-onto→𝑦 → 𝑓:(cf‘𝐴)–1-1→𝑦)
39		f1ss 6106	. . . . . . . . . . . 12 ⊢ ((𝑓:(cf‘𝐴)–1-1→𝑦 ∧ 𝑦 ⊆ 𝐴) → 𝑓:(cf‘𝐴)–1-1→𝐴)
40	39	ancoms 469	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑓:(cf‘𝐴)–1-1→𝑦) → 𝑓:(cf‘𝐴)–1-1→𝐴)
41	38, 40	sylan2 491	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑓:(cf‘𝐴)–1-1-onto→𝑦) → 𝑓:(cf‘𝐴)–1-1→𝐴)
42	41	adantlr 751	. . . . . . . . 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 1079	. . . . . . . 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 6144	. . . . . . . . . . . 12 ⊢ (𝑓:(cf‘𝐴)–1-1-onto→𝑦 → 𝑓:(cf‘𝐴)–onto→𝑦)
45		foelrn 6378	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘𝐴)–onto→𝑦 ∧ 𝑠 ∈ 𝑦) → ∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓‘𝑤))
46		sseq2 3627	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑠 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
47	46	biimpcd 239	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ 𝑠 → (𝑠 = (𝑓‘𝑤) → 𝑧 ⊆ (𝑓‘𝑤)))
48	47	reximdv 3016	. . . . . . . . . . . . . . 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 3031	. . . . . . . . . . . . 13 ⊢ (𝑓:(cf‘𝐴)–onto→𝑦 → (∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
51	50	ralimdv 2963	. . . . . . . . . . . 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 446	. . . . . . . . . 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 750	. . . . . . . . 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 1079	. . . . . . . 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 554	. . . . . . 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 1267	. . . . . 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 1846	. . . . 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 648	. . 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 1861	. 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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∩ cint 4475   class class class wbr 4653  dom cdm 5114  Oncon0 5723  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888   ≈ cen 7952  cardccrd 8761  cfccf 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-card 8765  df-cf 8767

This theorem is referenced by:  cfsmolem  9092  cfcoflem  9094  cfcof  9096  alephreg  9404