Theorem cfflb 10172

Description: If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 10171 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 6663	. . . . . . 7 ⊢ (𝑓:𝐵⟶𝐴 → ran 𝑓 ⊆ 𝐴)
2	1	adantr 480	. . . . . 6 ⊢ ((𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → ran 𝑓 ⊆ 𝐴)
3		ffn 6656	. . . . . . . . . . 11 ⊢ (𝑓:𝐵⟶𝐴 → 𝑓 Fn 𝐵)
4		fnfvelrn 7018	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑓‘𝑤) ∈ ran 𝑓)
5	3, 4	sylan 580	. . . . . . . . . 10 ⊢ ((𝑓:𝐵⟶𝐴 ∧ 𝑤 ∈ 𝐵) → (𝑓‘𝑤) ∈ ran 𝑓)
6		sseq2 3964	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑠 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
7	6	rspcev 3579	. . . . . . . . . 10 ⊢ (((𝑓‘𝑤) ∈ ran 𝑓 ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)
8	5, 7	sylan 580	. . . . . . . . 9 ⊢ (((𝑓:𝐵⟶𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)
9	8	rexlimdva2 3132	. . . . . . . 8 ⊢ (𝑓:𝐵⟶𝐴 → (∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤) → ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
10	9	ralimdv 3143	. . . . . . 7 ⊢ (𝑓:𝐵⟶𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤) → ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
11	10	imp 406	. . . . . 6 ⊢ ((𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)
12	2, 11	jca 511	. . . . 5 ⊢ ((𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
13		fvex 6839	. . . . . 6 ⊢ (card‘ran 𝑓) ∈ V
14		cfval 10160	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
15	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠))})
16	15	3ad2ant2 1134	. . . . . . . . 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 3442	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
18	17	rnex 7850	. . . . . . . . . . . . 13 ⊢ ran 𝑓 ∈ V
19		fveq2 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ran 𝑓 → (card‘𝑦) = (card‘ran 𝑓))
20	19	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ran 𝑓 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘ran 𝑓)))
21		sseq1 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ran 𝑓 → (𝑦 ⊆ 𝐴 ↔ ran 𝑓 ⊆ 𝐴))
22		rexeq 3286	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ran 𝑓 → (∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 ↔ ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
23	22	ralbidv 3152	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ran 𝑓 → (∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 ↔ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
24	21, 23	anbi12d 632	. . . . . . . . . . . . . 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 632	. . . . . . . . . . . . 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 3563	. . . . . . . . . . . 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 2711	. . . . . . . . . . . 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 234	. . . . . . . . . . 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 4916	. . . . . . . . . . 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 1131	. . . . . . . . 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 3972	. . . . . . . 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 1122	. . . . . . 7 ⊢ (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → (cf‘𝐴) ⊆ 𝑥))
34		sseq2 3964	. . . . . . 7 ⊢ (𝑥 = (card‘ran 𝑓) → ((cf‘𝐴) ⊆ 𝑥 ↔ (cf‘𝐴) ⊆ (card‘ran 𝑓)))
35	33, 34	sylibd 239	. . . . . 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 3512	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
37	12, 36	sylan2 593	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
38		cardidm 9874	. . . . . . 7 ⊢ (card‘(card‘ran 𝑓)) = (card‘ran 𝑓)
39		onss 7725	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
40	1, 39	sylan9ssr 3952	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ⊆ On)
41	40	3adant2 1131	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ⊆ On)
42		onssnum 9953	. . . . . . . . . . . 12 ⊢ ((ran 𝑓 ∈ V ∧ ran 𝑓 ⊆ On) → ran 𝑓 ∈ dom card)
43	18, 41, 42	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ∈ dom card)
44		cardid2 9868	. . . . . . . . . . 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 9864	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card)
47		dffn4 6746	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn 𝐵 ↔ 𝑓:𝐵–onto→ran 𝑓)
48	3, 47	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵⟶𝐴 → 𝑓:𝐵–onto→ran 𝑓)
49		fodomnum 9970	. . . . . . . . . . . . 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 406	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ≼ 𝐵)
52	51	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ≼ 𝐵)
53		endomtr 8944	. . . . . . . . . 10 ⊢ (((card‘ran 𝑓) ≈ ran 𝑓 ∧ ran 𝑓 ≼ 𝐵) → (card‘ran 𝑓) ≼ 𝐵)
54	45, 52, 53	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘ran 𝑓) ≼ 𝐵)
55		cardon 9859	. . . . . . . . . . . 12 ⊢ (card‘ran 𝑓) ∈ On
56		onenon 9864	. . . . . . . . . . . 12 ⊢ ((card‘ran 𝑓) ∈ On → (card‘ran 𝑓) ∈ dom card)
57	55, 56	ax-mp 5	. . . . . . . . . . 11 ⊢ (card‘ran 𝑓) ∈ dom card
58		carddom2 9892	. . . . . . . . . . 11 ⊢ (((card‘ran 𝑓) ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
59	57, 46, 58	sylancr 587	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
60	59	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
61	54, 60	mpbird 257	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘(card‘ran 𝑓)) ⊆ (card‘𝐵))
62		cardonle 9872	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (card‘𝐵) ⊆ 𝐵)
63	62	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘𝐵) ⊆ 𝐵)
64	61, 63	sstrd 3948	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘(card‘ran 𝑓)) ⊆ 𝐵)
65	38, 64	eqsstrrid 3977	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
66	65	3expa 1118	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑓:𝐵⟶𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
67	66	adantrr 717	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (card‘ran 𝑓) ⊆ 𝐵)
68	37, 67	sstrd 3948	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (cf‘𝐴) ⊆ 𝐵)
69	68	ex 412	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) ⊆ 𝐵))
70	69	exlimdv 1933	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3438   ⊆ wss 3905  ∩ cint 4899   class class class wbr 5095  dom cdm 5623  ran crn 5624  Oncon0 6311   Fn wfn 6481  ⟶wf 6482  –onto→wfo 6484  ‘cfv 6486   ≈ cen 8876   ≼ cdom 8877  cardccrd 9850  cfccf 9852

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-card 9854  df-cf 9856  df-acn 9857

This theorem is referenced by:  cfsmolem  10183  cfcoflem  10185  cfcof  10187  inar1  10688