Theorem trcfilu 24272

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 1137	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
2		simp2l 1201	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (CauFilu‘𝑈))
3		iscfilu 24266	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
4	3	biimpa 476	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
5	1, 2, 4	syl2anc 585	. . . 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 494	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (fBas‘𝑋))
7		simp3 1139	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
8		simp2r 1202	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → ¬ ∅ ∈ (𝐹 ↾t 𝐴))
9		trfbas2 23822	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹 ↾t 𝐴)))
10	9	biimpar 477	. . 3 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴))
11	6, 7, 8, 10	syl21anc 838	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴))
12	2	ad5antr 735	. . . . . . 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 480	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
14	13	elfvexd 6872	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑋 ∈ V)
15	7	adantr 480	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝐴 ⊆ 𝑋)
16	14, 15	ssexd 5262	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝐴 ∈ V)
17	16	ad4antr 733	. . . . . . 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 769	. . . . . . 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 17386	. . . . . . 7 ⊢ ((𝐹 ∈ (CauFilu‘𝑈) ∧ 𝐴 ∈ V ∧ 𝑎 ∈ 𝐹) → (𝑎 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
20	12, 17, 18, 19	syl3anc 1374	. . . . . 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 5782	. . . . . . 7 ⊢ ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) = ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴))
22		simpr 484	. . . . . . . . 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 4185	. . . . . . . 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 776	. . . . . . . 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 3960	. . . . . . 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 3962	. . . . . 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 5658	. . . . . . . 8 ⊢ (𝑏 = (𝑎 ∩ 𝐴) → (𝑏 × 𝑏) = ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)))
29	28	sseq1d 3954	. . . . . . 7 ⊢ (𝑏 = (𝑎 ∩ 𝐴) → ((𝑏 × 𝑏) ⊆ 𝑤 ↔ ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) ⊆ 𝑤))
30	29	rspcev 3565	. . . . . 6 ⊢ (((𝑎 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴) ∧ ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) ⊆ 𝑤) → ∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
31	20, 26, 30	syl2anc 585	. . . . 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 495	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
33	32	r19.21bi 3230	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
34	33	ad4ant13 752	. . . . 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 3146	. . . 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 7700	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
37		simpr 484	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
38		elrest 17385	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑣 ∈ 𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))))
39	38	biimpa 476	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑣 ∈ 𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
40	13, 36, 37, 39	syl21anc 838	. . . 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 3146	. . 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 3130	. 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 24208	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
44	1, 7, 43	syl2anc 585	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
45		iscfilu 24266	. . 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 714	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274   × cxp 5624  ‘cfv 6494  (class class class)co 7362   ↾_t crest 17378  fBascfbas 21336  UnifOncust 24179  CauFil_uccfilu 24264

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7937  df-2nd 7938  df-rest 17380  df-fbas 21345  df-ust 24180  df-cfilu 24265

This theorem is referenced by:  ucnextcn  24282