Theorem filssufilg 21994

Description: A filter is contained in some ultrafilter. This version of filssufil 21995 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 477	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → 𝒫 𝒫 𝑋 ∈ dom card)
2		rabss 3839	. . . . 5 ⊢ ({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ⊆ 𝒫 𝒫 𝑋 ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝑔 ∈ 𝒫 𝒫 𝑋))
3		filsspw 21934	. . . . . . 7 ⊢ (𝑔 ∈ (Fil‘𝑋) → 𝑔 ⊆ 𝒫 𝑋)
4		selpw 4322	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 𝒫 𝑋 ↔ 𝑔 ⊆ 𝒫 𝑋)
5	3, 4	sylibr 225	. . . . . 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 3074	. . . 4 ⊢ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ⊆ 𝒫 𝒫 𝑋
8		ssnum 9113	. . . 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 580	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∈ dom card)
10		ssid 3783	. . . . . . 7 ⊢ 𝐹 ⊆ 𝐹
11	10	jctr 520	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐹))
12		sseq2 3787	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝐹))
13	12	elrab 3519	. . . . . 6 ⊢ (𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐹))
14	11, 13	sylibr 225	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
15	14	ne0d 4086	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ≠ ∅)
16	15	adantr 472	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ≠ ∅)
17		simpr1 1248	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
18		ssrab 3840	. . . . . . . . . 10 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔 ∈ 𝑥 𝐹 ⊆ 𝑔))
19	17, 18	sylib 209	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔 ∈ 𝑥 𝐹 ⊆ 𝑔))
20	19	simpld 488	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ⊆ (Fil‘𝑋))
21		simpr2 1250	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝑥 ≠ ∅)
22		simpr3 1252	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → [⊊] Or 𝑥)
23		sorpssun 7142	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝑥 ∧ (𝑔 ∈ 𝑥 ∧ ℎ ∈ 𝑥)) → (𝑔 ∪ ℎ) ∈ 𝑥)
24	23	ralrimivva 3118	. . . . . . . . 9 ⊢ ( [⊊] Or 𝑥 → ∀𝑔 ∈ 𝑥 ∀ℎ ∈ 𝑥 (𝑔 ∪ ℎ) ∈ 𝑥)
25	22, 24	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∀𝑔 ∈ 𝑥 ∀ℎ ∈ 𝑥 (𝑔 ∪ ℎ) ∈ 𝑥)
26		filuni 21968	. . . . . . . 8 ⊢ ((𝑥 ⊆ (Fil‘𝑋) ∧ 𝑥 ≠ ∅ ∧ ∀𝑔 ∈ 𝑥 ∀ℎ ∈ 𝑥 (𝑔 ∪ ℎ) ∈ 𝑥) → ∪ 𝑥 ∈ (Fil‘𝑋))
27	20, 21, 25, 26	syl3anc 1490	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ (Fil‘𝑋))
28		n0 4095	. . . . . . . . 9 ⊢ (𝑥 ≠ ∅ ↔ ∃ℎ ℎ ∈ 𝑥)
29		ssel2 3756	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
30		sseq2 3787	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = ℎ → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ ℎ))
31	30	elrab 3519	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (ℎ ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ ℎ))
32	29, 31	sylib 209	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → (ℎ ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ ℎ))
33	32	simprd 489	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ℎ)
34		ssuni 4618	. . . . . . . . . . . 12 ⊢ ((𝐹 ⊆ ℎ ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ∪ 𝑥)
35	33, 34	sylancom 582	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ℎ ∈ 𝑥) → 𝐹 ⊆ ∪ 𝑥)
36	35	ex 401	. . . . . . . . . 10 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (ℎ ∈ 𝑥 → 𝐹 ⊆ ∪ 𝑥))
37	36	exlimdv 2028	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (∃ℎ ℎ ∈ 𝑥 → 𝐹 ⊆ ∪ 𝑥))
38	28, 37	syl5bi 233	. . . . . . . 8 ⊢ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} → (𝑥 ≠ ∅ → 𝐹 ⊆ ∪ 𝑥))
39	17, 21, 38	sylc 65	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → 𝐹 ⊆ ∪ 𝑥)
40		sseq2 3787	. . . . . . . 8 ⊢ (𝑔 = ∪ 𝑥 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ ∪ 𝑥))
41	40	elrab 3519	. . . . . . 7 ⊢ (∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (∪ 𝑥 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ ∪ 𝑥))
42	27, 39, 41	sylanbrc 578	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔})
43	42	ex 401	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}))
44	43	alrimiv 2022	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ 𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}))
45	44	adantr 472	. . 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}))
46		zornn0g 9580	. . 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)
47	9, 16, 45, 46	syl3anc 1490	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ)
48		sseq2 3787	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝑓))
49	48	elrab 3519	. . . 4 ⊢ (𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ↔ (𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
50	30	ralrab 3525	. . . 4 ⊢ (∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ ↔ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ))
51		simpll 783	. . . . . 6 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → 𝑓 ∈ (Fil‘𝑋))
52		sstr2 3768	. . . . . . . . . . 11 ⊢ (𝐹 ⊆ 𝑓 → (𝑓 ⊆ ℎ → 𝐹 ⊆ ℎ))
53	52	imim1d 82	. . . . . . . . . 10 ⊢ (𝐹 ⊆ 𝑓 → ((𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → (𝑓 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)))
54		df-pss 3748	. . . . . . . . . . . . 13 ⊢ (𝑓 ⊊ ℎ ↔ (𝑓 ⊆ ℎ ∧ 𝑓 ≠ ℎ))
55	54	simplbi2 494	. . . . . . . . . . . 12 ⊢ (𝑓 ⊆ ℎ → (𝑓 ≠ ℎ → 𝑓 ⊊ ℎ))
56	55	necon1bd 2955	. . . . . . . . . . 11 ⊢ (𝑓 ⊆ ℎ → (¬ 𝑓 ⊊ ℎ → 𝑓 = ℎ))
57	56	a2i 14	. . . . . . . . . 10 ⊢ ((𝑓 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → (𝑓 ⊆ ℎ → 𝑓 = ℎ))
58	53, 57	syl6 35	. . . . . . . . 9 ⊢ (𝐹 ⊆ 𝑓 → ((𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → (𝑓 ⊆ ℎ → 𝑓 = ℎ)))
59	58	ralimdv 3110	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝑓 → (∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ)))
60	59	imp 395	. . . . . . 7 ⊢ ((𝐹 ⊆ 𝑓 ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ))
61	60	adantll 705	. . . . . 6 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ))
62		isufil2 21991	. . . . . 6 ⊢ (𝑓 ∈ (UFil‘𝑋) ↔ (𝑓 ∈ (Fil‘𝑋) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝑓 ⊆ ℎ → 𝑓 = ℎ)))
63	51, 61, 62	sylanbrc 578	. . . . 5 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → 𝑓 ∈ (UFil‘𝑋))
64		simplr 785	. . . . 5 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → 𝐹 ⊆ 𝑓)
65	63, 64	jca 507	. . . 4 ⊢ (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) ∧ ∀ℎ ∈ (Fil‘𝑋)(𝐹 ⊆ ℎ → ¬ 𝑓 ⊊ ℎ)) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
66	49, 50, 65	syl2anb 591	. . 3 ⊢ ((𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ∧ ∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹 ⊆ 𝑓))
67	66	reximi2 3156	. 2 ⊢ (∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔}∀ℎ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹 ⊆ 𝑔} ¬ 𝑓 ⊊ ℎ → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)
68	47, 67	syl 17	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1107  ∀wal 1650  ∃wex 1874   ∈ wcel 2155   ≠ wne 2937  ∀wral 3055  ∃wrex 3056  {crab 3059   ∪ cun 3730   ⊆ wss 3732   ⊊ wpss 3733  ∅c0 4079  𝒫 cpw 4315  ∪ cuni 4594   Or wor 5197  dom cdm 5277  ‘cfv 6068   [⊊] crpss 7134  cardccrd 9012  Filcfil 21928  UFilcufil 21982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-rpss 7135  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-fin 8164  df-fi 8524  df-card 9016  df-cda 9243  df-fbas 20016  df-fg 20017  df-fil 21929  df-ufil 21984

This theorem is referenced by:  filssufil  21995  numufl  21998