Theorem filssufilg 24138

Description: A filter is contained in some ultrafilter. This version of filssufil 24139 contains the choice as a hypothesis (in the assumption that 𝒫 𝒫 𝑋 is well-orderable). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
filssufilg	⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)

Distinct variable groups:   𝑓,𝐹   𝑓,𝑋

Proof of Theorem filssufilg

Dummy variables 𝑔 ℎ 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → 𝒫 𝒫 𝑋 ∈ dom card)
2		rabss 4021	. . . . 5 ⊢ ({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ⊆ 𝒫 𝒫 𝑋 ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝑔 ∈ 𝒫 𝒫 𝑋))
3		filsspw 24078	. . . . . . 7 ⊢ (𝑔 ∈ (Fil‘𝑋) → 𝑔 ⊆ 𝒫 𝑋)
4		velpw 4565	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 𝒫 𝑋 ↔ 𝑔 ⊆ 𝒫 𝑋)
5	3, 4	sylibr 237	. . . . . 6 ⊢ (𝑔 ∈ (Fil‘𝑋) → 𝑔 ∈ 𝒫 𝒫 𝑋)
6	5	a1d 26	. . . . 5 ⊢ (𝑔 ∈ (Fil‘𝑋) → (𝐹 ⊆ 𝑔 → 𝑔 ∈ 𝒫 𝒫 𝑋))
7	2, 6	mprgbir 3085	. . . 4 ⊢ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ⊆ 𝒫 𝒫 𝑋
8		ssnum 10045	. . . 4 ⊢ ((𝒫 𝒫 𝑋 ∈ dom card ∧ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ⊆ 𝒫 𝒫 𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∈ dom card)
9	1, 7, 8	sylancl 598	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∈ dom card)
10		ssid 3956	. . . . . . 7 ⊢ 𝐹 ⊆ 𝐹
11	10	jctr 534	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐹))
12		sseq2 3960	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝐹))
13	12	elrab 3648	. . . . . 6 ⊢ (𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐹))
14	11, 13	sylibr 237	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
15	14	ne0d 4291	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ≠ ∅)
16	15	adantr 486	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ≠ ∅)
17		sseq2 3960	. . . . . . 7 ⊢ (𝑔 = ∪ 𝑥 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ ∪ 𝑥))
18		simpr1 1213	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
19		ssrab 4022	. . . . . . . . . 10 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔 ∈ 𝑥 𝐹 ⊆ 𝑔))
20	18, 19	sylib 221	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔 ∈ 𝑥 𝐹 ⊆ 𝑔))
21	20	simpld 500	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ⊆ (Fil‘𝑋))
22		simpr2 1214	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ≠ ∅)
23		simpr3 1215	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → [⊊] Or 𝑥)
24		sorpssun 7734	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝑥 ∧ (𝑔 ∈ 𝑥 ∧ ℎ ∈ 𝑥)) → (𝑔 ∪ ℎ) ∈ 𝑥)
25	24	ralrimivva 3207	. . . . . . . . 9 ⊢ ( [⊊] Or 𝑥 → ∀𝑔 ∈ 𝑥 ∀ℎ ∈ 𝑥 (𝑔 ∪ ℎ) ∈ 𝑥)
26	23, 25	syl 18	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∀𝑔 ∈ 𝑥 ∀ℎ ∈ 𝑥 (𝑔 ∪ ℎ) ∈ 𝑥)
27		filuni 24112	. . . . . . . 8 ⊢ ((𝑥 ⊆ (Fil‘𝑋) ∧ 𝑥 ≠ ∅ ∧ ∀𝑔 ∈ 𝑥 ∀ℎ ∈ 𝑥 (𝑔 ∪ ℎ) ∈ 𝑥) → ∪ 𝑥 ∈ (Fil‘𝑋))
28	21, 22, 26, 27	syl3anc 1398	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ (Fil‘𝑋))
29		n0 4303	. . . . . . . . 9 ⊢ (𝑥 ≠ ∅ ↔ ∃ℎ ℎ ∈ 𝑥)
30		ssel2 3929	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
31		sseq2 3960	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = ℎ → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ ℎ))
32	31	elrab 3648	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (ℎ ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ ℎ))
33	30, 32	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → (ℎ ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ ℎ))
34	33	simprd 501	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ℎ)
35		ssuni 4896	. . . . . . . . . . . 12 ⊢ ((𝐹 ⊆ ℎ ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ∪ 𝑥)
36	34, 35	sylancom 600	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ∪ 𝑥)
37	36	ex 418	. . . . . . . . . 10 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (ℎ ∈ 𝑥 → 𝐹 ⊆ ∪ 𝑥))
38	37	exlimdv 1966	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (∃ℎ ℎ ∈ 𝑥 → 𝐹 ⊆ ∪ 𝑥))
39	29, 38	biimtrid 245	. . . . . . . 8 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (𝑥 ≠ ∅ → 𝐹 ⊆ ∪ 𝑥))
40	18, 22, 39	sylc 66	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝐹 ⊆ ∪ 𝑥)
41	17, 28, 40	elrabd 3650	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
42	41	ex 418	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}))
43	42	alrimiv 1960	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}))
44	43	adantr 486	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}))
45		zornn0g 10510	. . 3 ⊢ (({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∈ dom card ∧ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ≠ ∅ ∧ ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})) → ∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ)
46	9, 16, 44, 45	syl3anc 1398	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ)
47		sseq2 3960	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝑓))
48	47	elrab 3648	. . . 4 ⊢ (𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
49	31	ralrab 3655	. . . 4 ⊢ (∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ ↔ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ))
50		simpll 779	. . . . . 6 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → 𝑓 ∈ (Fil‘𝑋))
51		sstr2 3941	. . . . . . . . . . 11 ⊢ (𝐹 ⊆ 𝑓 → (𝑓 ⊆ ℎ → 𝐹 ⊆ ℎ))
52	51	imim1d 83	. . . . . . . . . 10 ⊢ (𝐹 ⊆ 𝑓 → ((𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → (𝑓 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)))
53		df-pss 3922	. . . . . . . . . . . . 13 ⊢ (𝑓 ⊊ ℎ ↔ (𝑓 ⊆ ℎ ∧ 𝑓 ≠ ℎ))
54	53	simplbi2 506	. . . . . . . . . . . 12 ⊢ (𝑓 ⊆ ℎ → (𝑓 ≠ ℎ → 𝑓 ⊊ ℎ))
55	54	necon1bd 2975	. . . . . . . . . . 11 ⊢ (𝑓 ⊆ ℎ → (¬ 𝑓 ⊊ ℎ → 𝑓 = ℎ))
56	55	a2i 15	. . . . . . . . . 10 ⊢ ((𝑓 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → (𝑓 ⊆ ℎ → 𝑓 = ℎ))
57	52, 56	syl6 36	. . . . . . . . 9 ⊢ (𝐹 ⊆ 𝑓 → ((𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → (𝑓 ⊆ ℎ → 𝑓 = ℎ)))
58	57	ralimdv 3178	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝑓 → (∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ)))
59	58	imp 412	. . . . . . 7 ⊢ ((𝐹 ⊆ 𝑓 ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ))
60	59	adantll 727	. . . . . 6 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ))
61		isufil2 24135	. . . . . 6 ⊢ (𝑓 ∈ (UFil‘𝑋) ↔ (𝑓 ∈ (Fil‘𝑋) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ)))
62	50, 60, 61	sylanbrc 595	. . . . 5 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → 𝑓 ∈ (UFil‘𝑋))
63		simplr 781	. . . . 5 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → 𝐹 ⊆ 𝑓)
64	62, 63	jca 521	. . . 4 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
65	48, 49, 64	syl2anb 610	. . 3 ⊢ ((𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
66	65	reximi2 3097	. 2 ⊢ (∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)
67	46, 66	syl 18	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414   ∪ cun 3900   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4282  𝒫 cpw 4560  ∪ cuni 4870   Or wor 5566  dom cdm 5659  ‘cfv 6537   [⊊] crpss 7726  cardccrd 9943  Filcfil 24072  UFilcufil 24126

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-rpss 7727  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-er 8699  df-en 8956  df-dom 8957  df-fin 8959  df-fi 9384  df-dju 9909  df-card 9947  df-fbas 21583  df-fg 21584  df-fil 24073  df-ufil 24128

This theorem is used by:  filssufil  24139  numufl  24142