Theorem filssufilg 23854

Description: A filter is contained in some ultrafilter. This version of filssufil 23855 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 484	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → 𝒫 𝒫 𝑋 ∈ dom card)
2		rabss 4052	. . . . 5 ⊢ ({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ⊆ 𝒫 𝒫 𝑋 ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝑔 ∈ 𝒫 𝒫 𝑋))
3		filsspw 23794	. . . . . . 7 ⊢ (𝑔 ∈ (Fil‘𝑋) → 𝑔 ⊆ 𝒫 𝑋)
4		velpw 4585	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 𝒫 𝑋 ↔ 𝑔 ⊆ 𝒫 𝑋)
5	3, 4	sylibr 234	. . . . . 6 ⊢ (𝑔 ∈ (Fil‘𝑋) → 𝑔 ∈ 𝒫 𝒫 𝑋)
6	5	a1d 25	. . . . 5 ⊢ (𝑔 ∈ (Fil‘𝑋) → (𝐹 ⊆ 𝑔 → 𝑔 ∈ 𝒫 𝒫 𝑋))
7	2, 6	mprgbir 3059	. . . 4 ⊢ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ⊆ 𝒫 𝒫 𝑋
8		ssnum 10058	. . . 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 586	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∈ dom card)
10		ssid 3986	. . . . . . 7 ⊢ 𝐹 ⊆ 𝐹
11	10	jctr 524	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐹))
12		sseq2 3990	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝐹))
13	12	elrab 3676	. . . . . 6 ⊢ (𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐹))
14	11, 13	sylibr 234	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
15	14	ne0d 4322	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ≠ ∅)
16	15	adantr 480	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ≠ ∅)
17		sseq2 3990	. . . . . . 7 ⊢ (𝑔 = ∪ 𝑥 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ ∪ 𝑥))
18		simpr1 1195	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
19		ssrab 4053	. . . . . . . . . 10 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔 ∈ 𝑥 𝐹 ⊆ 𝑔))
20	18, 19	sylib 218	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔 ∈ 𝑥 𝐹 ⊆ 𝑔))
21	20	simpld 494	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ⊆ (Fil‘𝑋))
22		simpr2 1196	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ≠ ∅)
23		simpr3 1197	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → [⊊] Or 𝑥)
24		sorpssun 7729	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝑥 ∧ (𝑔 ∈ 𝑥 ∧ ℎ ∈ 𝑥)) → (𝑔 ∪ ℎ) ∈ 𝑥)
25	24	ralrimivva 3188	. . . . . . . . 9 ⊢ ( [⊊] Or 𝑥 → ∀𝑔 ∈ 𝑥 ∀ℎ ∈ 𝑥 (𝑔 ∪ ℎ) ∈ 𝑥)
26	23, 25	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∀𝑔 ∈ 𝑥 ∀ℎ ∈ 𝑥 (𝑔 ∪ ℎ) ∈ 𝑥)
27		filuni 23828	. . . . . . . 8 ⊢ ((𝑥 ⊆ (Fil‘𝑋) ∧ 𝑥 ≠ ∅ ∧ ∀𝑔 ∈ 𝑥 ∀ℎ ∈ 𝑥 (𝑔 ∪ ℎ) ∈ 𝑥) → ∪ 𝑥 ∈ (Fil‘𝑋))
28	21, 22, 26, 27	syl3anc 1373	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ (Fil‘𝑋))
29		n0 4333	. . . . . . . . 9 ⊢ (𝑥 ≠ ∅ ↔ ∃ℎ ℎ ∈ 𝑥)
30		ssel2 3958	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
31		sseq2 3990	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = ℎ → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ ℎ))
32	31	elrab 3676	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (ℎ ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ ℎ))
33	30, 32	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → (ℎ ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ ℎ))
34	33	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ℎ)
35		ssuni 4913	. . . . . . . . . . . 12 ⊢ ((𝐹 ⊆ ℎ ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ∪ 𝑥)
36	34, 35	sylancom 588	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ∪ 𝑥)
37	36	ex 412	. . . . . . . . . 10 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (ℎ ∈ 𝑥 → 𝐹 ⊆ ∪ 𝑥))
38	37	exlimdv 1933	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (∃ℎ ℎ ∈ 𝑥 → 𝐹 ⊆ ∪ 𝑥))
39	29, 38	biimtrid 242	. . . . . . . 8 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (𝑥 ≠ ∅ → 𝐹 ⊆ ∪ 𝑥))
40	18, 22, 39	sylc 65	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝐹 ⊆ ∪ 𝑥)
41	17, 28, 40	elrabd 3678	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
42	41	ex 412	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}))
43	42	alrimiv 1927	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}))
44	43	adantr 480	. . 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 10524	. . 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 1373	. 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 3990	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝑓))
48	47	elrab 3676	. . . 4 ⊢ (𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
49	31	ralrab 3682	. . . 4 ⊢ (∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ ↔ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ))
50		simpll 766	. . . . . 6 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → 𝑓 ∈ (Fil‘𝑋))
51		sstr2 3970	. . . . . . . . . . 11 ⊢ (𝐹 ⊆ 𝑓 → (𝑓 ⊆ ℎ → 𝐹 ⊆ ℎ))
52	51	imim1d 82	. . . . . . . . . 10 ⊢ (𝐹 ⊆ 𝑓 → ((𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → (𝑓 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)))
53		df-pss 3951	. . . . . . . . . . . . 13 ⊢ (𝑓 ⊊ ℎ ↔ (𝑓 ⊆ ℎ ∧ 𝑓 ≠ ℎ))
54	53	simplbi2 500	. . . . . . . . . . . 12 ⊢ (𝑓 ⊆ ℎ → (𝑓 ≠ ℎ → 𝑓 ⊊ ℎ))
55	54	necon1bd 2951	. . . . . . . . . . 11 ⊢ (𝑓 ⊆ ℎ → (¬ 𝑓 ⊊ ℎ → 𝑓 = ℎ))
56	55	a2i 14	. . . . . . . . . 10 ⊢ ((𝑓 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → (𝑓 ⊆ ℎ → 𝑓 = ℎ))
57	52, 56	syl6 35	. . . . . . . . 9 ⊢ (𝐹 ⊆ 𝑓 → ((𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → (𝑓 ⊆ ℎ → 𝑓 = ℎ)))
58	57	ralimdv 3155	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝑓 → (∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ)))
59	58	imp 406	. . . . . . 7 ⊢ ((𝐹 ⊆ 𝑓 ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ))
60	59	adantll 714	. . . . . 6 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ))
61		isufil2 23851	. . . . . 6 ⊢ (𝑓 ∈ (UFil‘𝑋) ↔ (𝑓 ∈ (Fil‘𝑋) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ)))
62	50, 60, 61	sylanbrc 583	. . . . 5 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → 𝑓 ∈ (UFil‘𝑋))
63		simplr 768	. . . . 5 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → 𝐹 ⊆ 𝑓)
64	62, 63	jca 511	. . . 4 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
65	48, 49, 64	syl2anb 598	. . 3 ⊢ ((𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
66	65	reximi2 3070	. 2 ⊢ (∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)
67	46, 66	syl 17	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1538  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3420   ∪ cun 3929   ⊆ wss 3931   ⊊ wpss 3932  ∅c0 4313  𝒫 cpw 4580  ∪ cuni 4888   Or wor 5565  dom cdm 5659  ‘cfv 6536   [⊊] crpss 7721  cardccrd 9954  Filcfil 23788  UFilcufil 23842

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  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 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-rpss 7722  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8724  df-en 8965  df-dom 8966  df-fin 8968  df-fi 9428  df-dju 9920  df-card 9958  df-fbas 21317  df-fg 21318  df-fil 23789  df-ufil 23844

This theorem is referenced by:  filssufil  23855  numufl  23858