Theorem filssufilg 22516

Description: A filter is contained in some ultrafilter. This version of filssufil 22517 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 488	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → 𝒫 𝒫 𝑋 ∈ dom card)
2		rabss 3999	. . . . 5 ⊢ ({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ⊆ 𝒫 𝒫 𝑋 ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝑔 ∈ 𝒫 𝒫 𝑋))
3		filsspw 22456	. . . . . . 7 ⊢ (𝑔 ∈ (Fil‘𝑋) → 𝑔 ⊆ 𝒫 𝑋)
4		velpw 4502	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 𝒫 𝑋 ↔ 𝑔 ⊆ 𝒫 𝑋)
5	3, 4	sylibr 237	. . . . . 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 3121	. . . 4 ⊢ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ⊆ 𝒫 𝒫 𝑋
8		ssnum 9450	. . . 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 589	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∈ dom card)
10		ssid 3937	. . . . . . 7 ⊢ 𝐹 ⊆ 𝐹
11	10	jctr 528	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐹))
12		sseq2 3941	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝐹))
13	12	elrab 3628	. . . . . 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 4251	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ≠ ∅)
16	15	adantr 484	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ≠ ∅)
17		sseq2 3941	. . . . . . 7 ⊢ (𝑔 = ∪ 𝑥 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ ∪ 𝑥))
18		simpr1 1191	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
19		ssrab 4000	. . . . . . . . . 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 498	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ⊆ (Fil‘𝑋))
22		simpr2 1192	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ≠ ∅)
23		simpr3 1193	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → [⊊] Or 𝑥)
24		sorpssun 7436	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝑥 ∧ (𝑔 ∈ 𝑥 ∧ ℎ ∈ 𝑥)) → (𝑔 ∪ ℎ) ∈ 𝑥)
25	24	ralrimivva 3156	. . . . . . . . 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 22490	. . . . . . . 8 ⊢ ((𝑥 ⊆ (Fil‘𝑋) ∧ 𝑥 ≠ ∅ ∧ ∀𝑔 ∈ 𝑥 ∀ℎ ∈ 𝑥 (𝑔 ∪ ℎ) ∈ 𝑥) → ∪ 𝑥 ∈ (Fil‘𝑋))
28	21, 22, 26, 27	syl3anc 1368	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ (Fil‘𝑋))
29		n0 4260	. . . . . . . . 9 ⊢ (𝑥 ≠ ∅ ↔ ∃ℎ ℎ ∈ 𝑥)
30		ssel2 3910	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
31		sseq2 3941	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = ℎ → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ ℎ))
32	31	elrab 3628	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ℎ)
35		ssuni 4825	. . . . . . . . . . . 12 ⊢ ((𝐹 ⊆ ℎ ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ∪ 𝑥)
36	34, 35	sylancom 591	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ∪ 𝑥)
37	36	ex 416	. . . . . . . . . 10 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (ℎ ∈ 𝑥 → 𝐹 ⊆ ∪ 𝑥))
38	37	exlimdv 1934	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (∃ℎ ℎ ∈ 𝑥 → 𝐹 ⊆ ∪ 𝑥))
39	29, 38	syl5bi 245	. . . . . . . 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 3630	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
42	41	ex 416	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}))
43	42	alrimiv 1928	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}))
44	43	adantr 484	. . 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 9916	. . 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 1368	. 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 3941	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝑓))
48	47	elrab 3628	. . . 4 ⊢ (𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
49	31	ralrab 3633	. . . 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 3922	. . . . . . . . . . 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 3900	. . . . . . . . . . . . 13 ⊢ (𝑓 ⊊ ℎ ↔ (𝑓 ⊆ ℎ ∧ 𝑓 ≠ ℎ))
54	53	simplbi2 504	. . . . . . . . . . . 12 ⊢ (𝑓 ⊆ ℎ → (𝑓 ≠ ℎ → 𝑓 ⊊ ℎ))
55	54	necon1bd 3005	. . . . . . . . . . 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 3145	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝑓 → (∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ)))
59	58	imp 410	. . . . . . 7 ⊢ ((𝐹 ⊆ 𝑓 ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ))
60	59	adantll 713	. . . . . 6 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ))
61		isufil2 22513	. . . . . 6 ⊢ (𝑓 ∈ (UFil‘𝑋) ↔ (𝑓 ∈ (Fil‘𝑋) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ)))
62	50, 60, 61	sylanbrc 586	. . . . 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 515	. . . 4 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
65	48, 49, 64	syl2anb 600	. . 3 ⊢ ((𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
66	65	reximi2 3207	. 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 399   ∧ w3a 1084  ∀wal 1536  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110   ∪ cun 3879   ⊆ wss 3881   ⊊ wpss 3882  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4800   Or wor 5437  dom cdm 5519  ‘cfv 6324   [⊊] crpss 7428  cardccrd 9348  Filcfil 22450  UFilcufil 22504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-rpss 7429  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-fin 8496  df-fi 8859  df-dju 9314  df-card 9352  df-fbas 20088  df-fg 20089  df-fil 22451  df-ufil 22506

This theorem is referenced by:  filssufil  22517  numufl  22520