Theorem restbas 22309

Description: A subspace topology basis is a basis. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
restbas	⊢ (𝐵 ∈ TopBases → (𝐵 ↾t 𝐴) ∈ TopBases)

Proof of Theorem restbas

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

Step	Hyp	Ref	Expression
1		elrest 17138	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝑎 ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐵 𝑎 = (𝑢 ∩ 𝐴)))
2		elrest 17138	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝑏 ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑣 ∈ 𝐵 𝑏 = (𝑣 ∩ 𝐴)))
3	1, 2	anbi12d 631	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → ((𝑎 ∈ (𝐵 ↾t 𝐴) ∧ 𝑏 ∈ (𝐵 ↾t 𝐴)) ↔ (∃𝑢 ∈ 𝐵 𝑎 = (𝑢 ∩ 𝐴) ∧ ∃𝑣 ∈ 𝐵 𝑏 = (𝑣 ∩ 𝐴))))
4		reeanv 3294	. . . . . 6 ⊢ (∃𝑢 ∈ 𝐵 ∃𝑣 ∈ 𝐵 (𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) ↔ (∃𝑢 ∈ 𝐵 𝑎 = (𝑢 ∩ 𝐴) ∧ ∃𝑣 ∈ 𝐵 𝑏 = (𝑣 ∩ 𝐴)))
5	3, 4	bitr4di 289	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → ((𝑎 ∈ (𝐵 ↾t 𝐴) ∧ 𝑏 ∈ (𝐵 ↾t 𝐴)) ↔ ∃𝑢 ∈ 𝐵 ∃𝑣 ∈ 𝐵 (𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴))))
6		simplll 772	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝐵 ∈ TopBases)
7		simplrl 774	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑢 ∈ 𝐵)
8		simplrr 775	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑣 ∈ 𝐵)
9		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴))
10	9	elin1d 4132	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑐 ∈ (𝑢 ∩ 𝑣))
11		basis2 22101	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ∈ 𝐵) ∧ (𝑣 ∈ 𝐵 ∧ 𝑐 ∈ (𝑢 ∩ 𝑣))) → ∃𝑧 ∈ 𝐵 (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))
12	6, 7, 8, 10, 11	syl22anc 836	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → ∃𝑧 ∈ 𝐵 (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))
13		simplll 772	. . . . . . . . . . . 12 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → (𝐵 ∈ TopBases ∧ 𝐴 ∈ V))
14	13	simpld 495	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝐵 ∈ TopBases)
15	13	simprd 496	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝐴 ∈ V)
16		simprl 768	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑧 ∈ 𝐵)
17		elrestr 17139	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (𝑧 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
18	14, 15, 16, 17	syl3anc 1370	. . . . . . . . . 10 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → (𝑧 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
19		simprrl 778	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑐 ∈ 𝑧)
20		simplr 766	. . . . . . . . . . . 12 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴))
21	20	elin2d 4133	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑐 ∈ 𝐴)
22	19, 21	elind 4128	. . . . . . . . . 10 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑐 ∈ (𝑧 ∩ 𝐴))
23		simprrr 779	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑧 ⊆ (𝑢 ∩ 𝑣))
24	23	ssrind 4169	. . . . . . . . . 10 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → (𝑧 ∩ 𝐴) ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))
25		eleq2 2827	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → (𝑐 ∈ 𝑤 ↔ 𝑐 ∈ (𝑧 ∩ 𝐴)))
26		sseq1 3946	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → (𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴) ↔ (𝑧 ∩ 𝐴) ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
27	25, 26	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → ((𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ↔ (𝑐 ∈ (𝑧 ∩ 𝐴) ∧ (𝑧 ∩ 𝐴) ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))))
28	27	rspcev 3561	. . . . . . . . . 10 ⊢ (((𝑧 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴) ∧ (𝑐 ∈ (𝑧 ∩ 𝐴) ∧ (𝑧 ∩ 𝐴) ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))) → ∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
29	18, 22, 24, 28	syl12anc 834	. . . . . . . . 9 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → ∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
30	12, 29	rexlimddv 3220	. . . . . . . 8 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → ∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
31	30	ralrimiva 3103	. . . . . . 7 ⊢ (((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ∀𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
32		ineq12 4141	. . . . . . . . 9 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑎 ∩ 𝑏) = ((𝑢 ∩ 𝐴) ∩ (𝑣 ∩ 𝐴)))
33		inindir 4161	. . . . . . . . 9 ⊢ ((𝑢 ∩ 𝑣) ∩ 𝐴) = ((𝑢 ∩ 𝐴) ∩ (𝑣 ∩ 𝐴))
34	32, 33	eqtr4di 2796	. . . . . . . 8 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑎 ∩ 𝑏) = ((𝑢 ∩ 𝑣) ∩ 𝐴))
35	34	sseq2d 3953	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑤 ⊆ (𝑎 ∩ 𝑏) ↔ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
36	35	anbi2d 629	. . . . . . . . 9 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → ((𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏)) ↔ (𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))))
37	36	rexbidv 3226	. . . . . . . 8 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏)) ↔ ∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))))
38	34, 37	raleqbidv 3336	. . . . . . 7 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (∀𝑐 ∈ (𝑎 ∩ 𝑏)∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏)) ↔ ∀𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))))
39	31, 38	syl5ibrcom 246	. . . . . 6 ⊢ (((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → ∀𝑐 ∈ (𝑎 ∩ 𝑏)∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏))))
40	39	rexlimdvva 3223	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (∃𝑢 ∈ 𝐵 ∃𝑣 ∈ 𝐵 (𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → ∀𝑐 ∈ (𝑎 ∩ 𝑏)∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏))))
41	5, 40	sylbid 239	. . . 4 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → ((𝑎 ∈ (𝐵 ↾t 𝐴) ∧ 𝑏 ∈ (𝐵 ↾t 𝐴)) → ∀𝑐 ∈ (𝑎 ∩ 𝑏)∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏))))
42	41	ralrimivv 3122	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → ∀𝑎 ∈ (𝐵 ↾t 𝐴)∀𝑏 ∈ (𝐵 ↾t 𝐴)∀𝑐 ∈ (𝑎 ∩ 𝑏)∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏)))
43		ovex 7308	. . . 4 ⊢ (𝐵 ↾t 𝐴) ∈ V
44		isbasis2g 22098	. . . 4 ⊢ ((𝐵 ↾t 𝐴) ∈ V → ((𝐵 ↾t 𝐴) ∈ TopBases ↔ ∀𝑎 ∈ (𝐵 ↾t 𝐴)∀𝑏 ∈ (𝐵 ↾t 𝐴)∀𝑐 ∈ (𝑎 ∩ 𝑏)∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏))))
45	43, 44	ax-mp 5	. . 3 ⊢ ((𝐵 ↾t 𝐴) ∈ TopBases ↔ ∀𝑎 ∈ (𝐵 ↾t 𝐴)∀𝑏 ∈ (𝐵 ↾t 𝐴)∀𝑐 ∈ (𝑎 ∩ 𝑏)∃𝑤 ∈ (𝐵 ↾t 𝐴)(𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏)))
46	42, 45	sylibr 233	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ TopBases)
47		relxp 5607	. . . . . 6 ⊢ Rel (V × V)
48		restfn 17135	. . . . . . . 8 ⊢ ↾t Fn (V × V)
49		fndm 6536	. . . . . . . 8 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V))
50	48, 49	ax-mp 5	. . . . . . 7 ⊢ dom ↾t = (V × V)
51	50	releqi 5688	. . . . . 6 ⊢ (Rel dom ↾t ↔ Rel (V × V))
52	47, 51	mpbir 230	. . . . 5 ⊢ Rel dom ↾t
53	52	ovprc2 7315	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵 ↾t 𝐴) = ∅)
54	53	adantl 482	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ ¬ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) = ∅)
55		fi0 9179	. . . 4 ⊢ (fi‘∅) = ∅
56		fibas 22127	. . . 4 ⊢ (fi‘∅) ∈ TopBases
57	55, 56	eqeltrri 2836	. . 3 ⊢ ∅ ∈ TopBases
58	54, 57	eqeltrdi 2847	. 2 ⊢ ((𝐵 ∈ TopBases ∧ ¬ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ TopBases)
59	46, 58	pm2.61dan 810	1 ⊢ (𝐵 ∈ TopBases → (𝐵 ↾t 𝐴) ∈ TopBases)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256   × cxp 5587  dom cdm 5589  Rel wrel 5594   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  ficfi 9169   ↾_t crest 17131  TopBasesctb 22095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-en 8734  df-fin 8737  df-fi 9170  df-rest 17133  df-bases 22096

This theorem is referenced by:  resttop  22311  2ndcrest  22605