Theorem trcfilu 22905

Description: Condition for the trace of a Cauchy filter base to be a Cauchy filter base for the restricted uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)

Assertion

Ref	Expression
trcfilu	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴))))

Proof of Theorem trcfilu

Dummy variables 𝑎 𝑏 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1132	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
2		simp2l 1195	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (CauFilu‘𝑈))
3		iscfilu 22899	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
4	3	biimpa 479	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
5	1, 2, 4	syl2anc 586	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
6	5	simpld 497	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (fBas‘𝑋))
7		simp3 1134	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
8		simp2r 1196	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → ¬ ∅ ∈ (𝐹 ↾t 𝐴))
9		trfbas2 22453	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹 ↾t 𝐴)))
10	9	biimpar 480	. . 3 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴))
11	6, 7, 8, 10	syl21anc 835	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴))
12	2	ad5antr 732	. . . . . . 7 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝐹 ∈ (CauFilu‘𝑈))
13	1	adantr 483	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
14	13	elfvexd 6706	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑋 ∈ V)
15	7	adantr 483	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝐴 ⊆ 𝑋)
16	14, 15	ssexd 5230	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝐴 ∈ V)
17	16	ad4antr 730	. . . . . . 7 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝐴 ∈ V)
18		simplr 767	. . . . . . 7 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝑎 ∈ 𝐹)
19		elrestr 16704	. . . . . . 7 ⊢ ((𝐹 ∈ (CauFilu‘𝑈) ∧ 𝐴 ∈ V ∧ 𝑎 ∈ 𝐹) → (𝑎 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
20	12, 17, 18, 19	syl3anc 1367	. . . . . 6 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → (𝑎 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
21		inxp 5705	. . . . . . 7 ⊢ ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) = ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴))
22		simpr 487	. . . . . . . . 9 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → (𝑎 × 𝑎) ⊆ 𝑣)
23	22	ssrind 4214	. . . . . . . 8 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
24		simpllr 774	. . . . . . . 8 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
25	23, 24	sseqtrrd 4010	. . . . . . 7 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) ⊆ 𝑤)
26	21, 25	eqsstrrid 4018	. . . . . 6 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) ⊆ 𝑤)
27		id 22	. . . . . . . . 9 ⊢ (𝑏 = (𝑎 ∩ 𝐴) → 𝑏 = (𝑎 ∩ 𝐴))
28	27	sqxpeqd 5589	. . . . . . . 8 ⊢ (𝑏 = (𝑎 ∩ 𝐴) → (𝑏 × 𝑏) = ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)))
29	28	sseq1d 4000	. . . . . . 7 ⊢ (𝑏 = (𝑎 ∩ 𝐴) → ((𝑏 × 𝑏) ⊆ 𝑤 ↔ ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) ⊆ 𝑤))
30	29	rspcev 3625	. . . . . 6 ⊢ (((𝑎 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴) ∧ ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) ⊆ 𝑤) → ∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
31	20, 26, 30	syl2anc 586	. . . . 5 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
32	5	simprd 498	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
33	32	r19.21bi 3210	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
34	33	ad4ant13 749	. . . . 5 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
35	31, 34	r19.29a 3291	. . . 4 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
36	16, 16	xpexd 7476	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
37		simpr 487	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
38		elrest 16703	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑣 ∈ 𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))))
39	38	biimpa 479	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑣 ∈ 𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
40	13, 36, 37, 39	syl21anc 835	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑣 ∈ 𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
41	35, 40	r19.29a 3291	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
42	41	ralrimiva 3184	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
43		trust 22840	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
44	1, 7, 43	syl2anc 586	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
45		iscfilu 22899	. . 3 ⊢ ((𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → ((𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ∧ ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)))
46	44, 45	syl 17	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ∧ ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)))
47	11, 42, 46	mpbir2and 711	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293   × cxp 5555  ‘cfv 6357  (class class class)co 7158   ↾_t crest 16696  fBascfbas 20535  UnifOncust 22810  CauFil_uccfilu 22897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-rest 16698  df-fbas 20544  df-ust 22811  df-cfilu 22898

This theorem is referenced by:  ucnextcn  22915