Theorem cfflb 9534

Description: If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 9533 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 6395	. . . . . . 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 6389	. . . . . . . . . . 11 ⊢ (𝑓:𝐵⟶𝐴 → 𝑓 Fn 𝐵)
4		fnfvelrn 6720	. . . . . . . . . . 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 3920	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑠 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
7	6	rspcev 3561	. . . . . . . . . 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 3252	. . . . . . . 8 ⊢ (𝑓:𝐵⟶𝐴 → (∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤) → ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
10	9	ralimdv 3147	. . . . . . 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 512	. . . . 5 ⊢ ((𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
13		fvex 6558	. . . . . 6 ⊢ (card‘ran 𝑓) ∈ V
14		cfval 9522	. . . . . . . . . . 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 1127	. . . . . . . . 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 3443	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
18	17	rnex 7480	. . . . . . . . . . . . 13 ⊢ ran 𝑓 ∈ V
19		fveq2 6545	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ran 𝑓 → (card‘𝑦) = (card‘ran 𝑓))
20	19	eqeq2d 2807	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ran 𝑓 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘ran 𝑓)))
21		sseq1 3919	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ran 𝑓 → (𝑦 ⊆ 𝐴 ↔ ran 𝑓 ⊆ 𝐴))
22		rexeq 3368	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ran 𝑓 → (∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 ↔ ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
23	22	ralbidv 3166	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ran 𝑓 → (∀𝑧 ∈ 𝐴 ∃𝑠 ∈ 𝑦 𝑧 ⊆ 𝑠 ↔ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠))
24	21, 23	anbi12d 630	. . . . . . . . . . . . . 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 630	. . . . . . . . . . . . 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 2781	. . . . . . . . . . . 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 4803	. . . . . . . . . . 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 1124	. . . . . . . . 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 3932	. . . . . . . 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 1115	. . . . . . 7 ⊢ (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → (cf‘𝐴) ⊆ 𝑥))
34		sseq2 3920	. . . . . . 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 3529	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑠 ∈ ran 𝑓 𝑧 ⊆ 𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
37	12, 36	sylan2 592	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
38		cardidm 9241	. . . . . . 7 ⊢ (card‘(card‘ran 𝑓)) = (card‘ran 𝑓)
39		onss 7368	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
40	1, 39	sylan9ssr 3909	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ⊆ On)
41	40	3adant2 1124	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ⊆ On)
42		onssnum 9319	. . . . . . . . . . . 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 9235	. . . . . . . . . . 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 9231	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card)
47		dffn4 6471	. . . . . . . . . . . . . 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 9336	. . . . . . . . . . . . 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 1123	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → ran 𝑓 ≼ 𝐵)
53		endomtr 8422	. . . . . . . . . 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 9226	. . . . . . . . . . . 12 ⊢ (card‘ran 𝑓) ∈ On
56		onenon 9231	. . . . . . . . . . . 12 ⊢ ((card‘ran 𝑓) ∈ On → (card‘ran 𝑓) ∈ dom card)
57	55, 56	ax-mp 5	. . . . . . . . . . 11 ⊢ (card‘ran 𝑓) ∈ dom card
58		carddom2 9259	. . . . . . . . . . 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 1127	. . . . . . . . 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 9239	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (card‘𝐵) ⊆ 𝐵)
63	62	3ad2ant2 1127	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘𝐵) ⊆ 𝐵)
64	61, 63	sstrd 3905	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘(card‘ran 𝑓)) ⊆ 𝐵)
65	38, 64	eqsstrrid 3943	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵⟶𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
66	65	3expa 1111	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑓:𝐵⟶𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
67	66	adantrr 713	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (card‘ran 𝑓) ⊆ 𝐵)
68	37, 67	sstrd 3905	. . 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 1915	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 1080   = wceq 1525  ∃wex 1765   ∈ wcel 2083  {cab 2777  ∀wral 3107  ∃wrex 3108  Vcvv 3440   ⊆ wss 3865  ∩ cint 4788   class class class wbr 4968  dom cdm 5450  ran crn 5451  Oncon0 6073   Fn wfn 6227  ⟶wf 6228  –onto→wfo 6230  ‘cfv 6232   ≈ cen 8361   ≼ cdom 8362  cardccrd 9217  cfccf 9219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-er 8146  df-map 8265  df-en 8365  df-dom 8366  df-sdom 8367  df-card 9221  df-cf 9223  df-acn 9224

This theorem is referenced by:  cfsmolem  9545  cfcoflem  9547  cfcof  9549  inar1  10050