Theorem trcfilu 24459

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 1154	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
2		simp2l 1218	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (CauFilu‘𝑈))
3		iscfilu 24453	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
4	3	biimpa 481	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
5	1, 2, 4	syl2anc 595	. . . 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 499	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (fBas‘𝑋))
7		simp3 1156	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
8		simp2r 1219	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → ¬ ∅ ∈ (𝐹 ↾t 𝐴))
9		trfbas2 24009	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹 ↾t 𝐴)))
10	9	biimpar 482	. . 3 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴))
11	6, 7, 8, 10	syl21anc 850	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴))
12	2	ad5antr 746	. . . . . . 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 485	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
14	13	elfvexd 6917	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑋 ∈ V)
15	7	adantr 485	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝐴 ⊆ 𝑋)
16	14, 15	ssexd 5295	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝐴 ∈ V)
17	16	ad4antr 744	. . . . . . 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 780	. . . . . . 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 17485	. . . . . . 7 ⊢ ((𝐹 ∈ (CauFilu‘𝑈) ∧ 𝐴 ∈ V ∧ 𝑎 ∈ 𝐹) → (𝑎 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
20	12, 17, 18, 19	syl3anc 1398	. . . . . 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 5818	. . . . . . 7 ⊢ ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) = ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴))
22		simpr 489	. . . . . . . . 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 4196	. . . . . . . 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 787	. . . . . . . 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 3974	. . . . . . 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 3976	. . . . . 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 23	. . . . . . . . 9 ⊢ (𝑏 = (𝑎 ∩ 𝐴) → 𝑏 = (𝑎 ∩ 𝐴))
28	27	sqxpeqd 5693	. . . . . . . 8 ⊢ (𝑏 = (𝑎 ∩ 𝐴) → (𝑏 × 𝑏) = ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)))
29	28	sseq1d 3968	. . . . . . 7 ⊢ (𝑏 = (𝑎 ∩ 𝐴) → ((𝑏 × 𝑏) ⊆ 𝑤 ↔ ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) ⊆ 𝑤))
30	29	rspcev 3581	. . . . . 6 ⊢ (((𝑎 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴) ∧ ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) ⊆ 𝑤) → ∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
31	20, 26, 30	syl2anc 595	. . . . 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 500	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
33	32	r19.21bi 3257	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
34	33	ad4ant13 763	. . . . 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 3173	. . . 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 7746	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
37		simpr 489	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
38		elrest 17484	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑣 ∈ 𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))))
39	38	biimpa 481	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑣 ∈ 𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
40	13, 36, 37, 39	syl21anc 850	. . . 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 3173	. . 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 3157	. 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 24395	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
44	1, 7, 43	syl2anc 595	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
45		iscfilu 24453	. . 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 18	. 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 725	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286   × cxp 5659  ‘cfv 6536  (class class class)co 7410   ↾_t crest 17477  fBascfbas 21519  UnifOncust 24366  CauFil_uccfilu 24451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-rest 17479  df-fbas 21528  df-ust 24367  df-cfilu 24452

This theorem is used by:  ucnextcn  24469