Theorem cfflb 10179

Description: If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 10178 motivate the picture of (cf‘𝐴) as the greatest lower bound of the domain of cofinal maps into 𝐴. (Contributed by Mario Carneiro, 28-Feb-2013.)

Assertion

Ref	Expression
cfflb	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) ⊆ 𝐵))

Distinct variable groups:   𝐴,𝑓,𝑤,𝑧   𝐵,𝑓,𝑤,𝑧

Proof of Theorem cfflb

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

Step	Hyp	Ref	Expression
1		frn 6669	. . . . . . 7 ⊢ (𝑓:𝐵⟶𝐴 → ran 𝑓 ⊆ 𝐴)
2	1	adantr 481	. . . . . 6 ⊢ ((𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → ran 𝑓 ⊆ 𝐴)
3		ffn 6662	. . . . . . . . . . 11 ⊢ (𝑓:𝐵⟶𝐴 → 𝑓 Fn 𝐵)
4		fnfvelrn 7028	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑓‘𝑤) ∈ ran 𝑓)
5	3, 4	sylan 586	. . . . . . . . . 10 ⊢ ((𝑓:𝐵⟶𝐴 ∧ 𝑤 ∈ 𝐵) → (𝑓‘𝑤) ∈ ran 𝑓)
6		sseq2 3948	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑠 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
7	6	rspcev 3567	. . . . . . . . . 10 ⊢ (((𝑓‘𝑤) ∈ ran 𝑓 ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)
8	5, 7	sylan 586	. . . . . . . . 9 ⊢ (((𝑓:𝐵⟶𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)
9	8	rexlimdva2 3143	. . . . . . . 8 ⊢ (𝑓:𝐵⟶𝐴 → (∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤) → ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
10	9	ralimdv 3154	. . . . . . 7 ⊢ (𝑓:𝐵⟶𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤) → ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
11	10	imp 407	. . . . . 6 ⊢ ((𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)
12	2, 11	jca 516	. . . . 5 ⊢ ((𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
13		fvex 6847	. . . . . 6 ⊢ (card‘ran 𝑓) ∈ V
14		cfval 10167	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
15	14	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
16	15	3ad2ant2 1140	. . . . . . . . 9 ⊢ ((𝑥 = (card‘ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
17		vex 3436	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
18	17	rnex 7857	. . . . . . . . . . . . 13 ⊢ ran 𝑓 ∈ V
19		fveq2 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ran 𝑓 → (card‘𝑦) = (card‘ran 𝑓))
20	19	eqeq2d 2751	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ran 𝑓 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘ran 𝑓)))
21		sseq1 3947	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ran 𝑓 → (𝑦 ⊆ 𝐴 ↔ ran 𝑓 ⊆ 𝐴))
22		rexeq 3294	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ran 𝑓 → (∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 ↔ ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
23	22	ralbidv 3163	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ran 𝑓 → (∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 ↔ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
24	21, 23	anbi12d 638	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ran 𝑓 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠) ↔ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)))
25	20, 24	anbi12d 638	. . . . . . . . . . . . 13 ⊢ (𝑦 = ran 𝑓 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)) ↔ (𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))))
26	18, 25	spcev 3551	. . . . . . . . . . . 12 ⊢ ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
27		abid 2722	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ↔ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠)))
28	26, 27	sylibr 235	. . . . . . . . . . 11 ⊢ ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → 𝑥 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
29		intss1 4900	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {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)
30	28, 29	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ⊆ 𝑥)
31	30	3adant2 1137	. . . . . . . . 9 ⊢ ((𝑥 = (card‘ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))} ⊆ 𝑥)
32	16, 31	eqsstrd 3956	. . . . . . . 8 ⊢ ((𝑥 = (card‘ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → (cf‘𝐴) ⊆ 𝑥)
33	32	3expib 1128	. . . . . . 7 ⊢ (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → (cf‘𝐴) ⊆ 𝑥))
34		sseq2 3948	. . . . . . 7 ⊢ (𝑥 = (card‘ran 𝑓) → ((cf‘𝐴) ⊆ 𝑥 ↔ (cf‘𝐴) ⊆ (card‘ran 𝑓)))
35	33, 34	sylibd 240	. . . . . 6 ⊢ (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓)))
36	13, 35	vtocle 3503	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
37	12, 36	sylan2 599	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
38		cardidm 9881	. . . . . . 7 ⊢ (card‘(card‘ran 𝑓)) = (card‘ran 𝑓)
39		onss 7735	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
40	1, 39	sylan9ssr 3936	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ⊆ On)
41	40	3adant2 1137	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ⊆ On)
42		onssnum 9960	. . . . . . . . . . . 12 ⊢ ((ran 𝑓 ∈ V ∧ ran 𝑓 ⊆ On) → ran 𝑓 ∈ dom card)
43	18, 41, 42	sylancr 593	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ∈ dom card)
44		cardid2 9875	. . . . . . . . . . 11 ⊢ (ran 𝑓 ∈ dom card → (card‘ran 𝑓) ≈ ran 𝑓)
45	43, 44	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘ran 𝑓) ≈ ran 𝑓)
46		onenon 9871	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card)
47		dffn4 6752	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn 𝐵 ↔ 𝑓:𝐵–onto→ran 𝑓)
48	3, 47	sylib 219	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵⟶𝐴 → 𝑓:𝐵–onto→ran 𝑓)
49		fodomnum 9977	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ dom card → (𝑓:𝐵–onto→ran 𝑓 → ran 𝑓 ≼ 𝐵))
50	46, 48, 49	syl2im 40	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → (𝑓:𝐵⟶𝐴 → ran 𝑓 ≼ 𝐵))
51	50	imp 407	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ≼ 𝐵)
52	51	3adant1 1136	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ≼ 𝐵)
53		endomtr 8956	. . . . . . . . . 10 ⊢ (((card‘ran 𝑓) ≈ ran 𝑓 ∧ ran 𝑓 ≼ 𝐵) → (card‘ran 𝑓) ≼ 𝐵)
54	45, 52, 53	syl2anc 590	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘ran 𝑓) ≼ 𝐵)
55		cardon 9866	. . . . . . . . . . . 12 ⊢ (card‘ran 𝑓) ∈ On
56		onenon 9871	. . . . . . . . . . . 12 ⊢ ((card‘ran 𝑓) ∈ On → (card‘ran 𝑓) ∈ dom card)
57	55, 56	ax-mp 5	. . . . . . . . . . 11 ⊢ (card‘ran 𝑓) ∈ dom card
58		carddom2 9899	. . . . . . . . . . 11 ⊢ (((card‘ran 𝑓) ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
59	57, 46, 58	sylancr 593	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
60	59	3ad2ant2 1140	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
61	54, 60	mpbird 258	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘(card‘ran 𝑓)) ⊆ (card‘𝐵))
62		cardonle 9879	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (card‘𝐵) ⊆ 𝐵)
63	62	3ad2ant2 1140	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘𝐵) ⊆ 𝐵)
64	61, 63	sstrd 3932	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘(card‘ran 𝑓)) ⊆ 𝐵)
65	38, 64	eqsstrrid 3961	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
66	65	3expa 1124	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑓:𝐵⟶𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
67	66	adantrr 723	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (card‘ran 𝑓) ⊆ 𝐵)
68	37, 67	sstrd 3932	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (cf‘𝐴) ⊆ 𝐵)
69	68	ex 413	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) ⊆ 𝐵))
70	69	exlimdv 1940	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2718  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ⊆ wss 3890  ∩ cint 4884   class class class wbr 5079  dom cdm 5625  ran crn 5626  Oncon0 6317   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492   ≈ cen 8887   ≼ cdom 8888  cardccrd 9857  cfccf 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-card 9861  df-cf 9863  df-acn 9864

This theorem is referenced by:  cfsmolem  10190  cfcoflem  10192  cfcof  10194  inar1  10696