Theorem cff1 10194

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 10183	. . . 4 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
2		cardon 9880	. . . . . . . . 9 ⊢ (card‘𝑦) ∈ On
3		eleq1 2825	. . . . . . . . 9 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
4	2, 3	mpbiri 257	. . . . . . . 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 1933	. . . . . 6 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑥 ∈ On)
7	6	abssi 4027	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ⊆ On
8		cflem 10182	. . . . . 6 ⊢ (𝐴 ∈ On → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
9		abn0 4340	. . . . . 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 233	. . . . 5 ⊢ (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ≠ ∅)
11		onint 7725	. . . . 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 2838	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
14		fvex 6855	. . . 4 ⊢ (cf‘𝐴) ∈ V
15		eqeq1 2740	. . . . . 6 ⊢ (𝑥 = (cf‘𝐴) → (𝑥 = (card‘𝑦) ↔ (cf‘𝐴) = (card‘𝑦)))
16	15	anbi1d 630	. . . . 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 1924	. . . 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 3630	. . 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 217	. 2 ⊢ (𝐴 ∈ On → ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
20		simplr 767	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → (cf‘𝐴) = (card‘𝑦))
21		onss 7719	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
22		sstr 3952	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑦 ⊆ On)
23	21, 22	sylan2 593	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ∈ On) → 𝑦 ⊆ On)
24	23	ancoms 459	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
25	24	ad2ant2r 745	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) → 𝑦 ⊆ On)
26		vex 3449	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
27		onssnum 9976	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
28	26, 27	mpan 688	. . . . . . . . . 10 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
29		cardid2 9889	. . . . . . . . . 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 5108	. . . . . . . . 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 256	. . . . . . 7 ⊢ (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (cf‘𝐴) ≈ 𝑦)
35		bren 8893	. . . . . . 7 ⊢ ((cf‘𝐴) ≈ 𝑦 ↔ ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto→𝑦)
36	34, 35	sylib 217	. . . . . 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 6783	. . . . . . . . . . 11 ⊢ (𝑓:(cf‘𝐴)–1-1-onto→𝑦 → 𝑓:(cf‘𝐴)–1-1→𝑦)
39		f1ss 6744	. . . . . . . . . . . 12 ⊢ ((𝑓:(cf‘𝐴)–1-1→𝑦 ∧ 𝑦 ⊆ 𝐴) → 𝑓:(cf‘𝐴)–1-1→𝐴)
40	39	ancoms 459	. . . . . . . . . . 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 713	. . . . . . . . 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 6791	. . . . . . . . . . . 12 ⊢ (𝑓:(cf‘𝐴)–1-1-onto→𝑦 → 𝑓:(cf‘𝐴)–onto→𝑦)
45		foelrn 7056	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘𝐴)–onto→𝑦 ∧ 𝑠 ∈ 𝑦) → ∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓‘𝑤))
46		sseq2 3970	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑠 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
47	46	biimpcd 248	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ 𝑠 → (𝑠 = (𝑓‘𝑤) → 𝑧 ⊆ (𝑓‘𝑤)))
48	47	reximdv 3167	. . . . . . . . . . . . . . 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 3152	. . . . . . . . . . . . 13 ⊢ (𝑓:(cf‘𝐴)–onto→𝑦 → (∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
51	50	ralimdv 3166	. . . . . . . . . . . 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 408	. . . . . . . . . 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 712	. . . . . . . . 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 512	. . . . . . 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 1920	. . . . 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 458	. . 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 1936	. 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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2713   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  ∩ cint 4907   class class class wbr 5105  dom cdm 5633  Oncon0 6317  –1-1→wf1 6493  –onto→wfo 6494  –1-1-onto→wf1o 6495  ‘cfv 6496   ≈ cen 8880  cardccrd 9871  cfccf 9873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-er 8648  df-en 8884  df-dom 8885  df-card 9875  df-cf 9877

This theorem is referenced by:  cfsmolem  10206  cfcoflem  10208  cfcof  10210  alephreg  10518