Theorem restbas 23083

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 17341	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝑎 ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐵 𝑎 = (𝑢 ∩ 𝐴)))
2		elrest 17341	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝑏 ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑣 ∈ 𝐵 𝑏 = (𝑣 ∩ 𝐴)))
3	1, 2	anbi12d 632	. . . . . 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 3206	. . . . . 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 774	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝐵 ∈ TopBases)
7		simplrl 776	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑢 ∈ 𝐵)
8		simplrr 777	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑣 ∈ 𝐵)
9		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴))
10	9	elin1d 4155	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑐 ∈ (𝑢 ∩ 𝑣))
11		basis2 22876	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ∈ 𝐵) ∧ (𝑣 ∈ 𝐵 ∧ 𝑐 ∈ (𝑢 ∩ 𝑣))) → ∃𝑧 ∈ 𝐵 (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))
12	6, 7, 8, 10, 11	syl22anc 838	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → ∃𝑧 ∈ 𝐵 (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))
13		simplll 774	. . . . . . . . . . . 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 494	. . . . . . . . . . 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 495	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝐴 ∈ V)
16		simprl 770	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑧 ∈ 𝐵)
17		elrestr 17342	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (𝑧 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
18	14, 15, 16, 17	syl3anc 1373	. . . . . . . . . 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 780	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑐 ∈ 𝑧)
20		simplr 768	. . . . . . . . . . . 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 4156	. . . . . . . . . . 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 4151	. . . . . . . . . 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 781	. . . . . . . . . . 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 4195	. . . . . . . . . 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 2822	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → (𝑐 ∈ 𝑤 ↔ 𝑐 ∈ (𝑧 ∩ 𝐴)))
26		sseq1 3957	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → (𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴) ↔ (𝑧 ∩ 𝐴) ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
27	25, 26	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → ((𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ↔ (𝑐 ∈ (𝑧 ∩ 𝐴) ∧ (𝑧 ∩ 𝐴) ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))))
28	27	rspcev 3574	. . . . . . . . . 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 836	. . . . . . . . 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 3141	. . . . . . . 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 3126	. . . . . . 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 4166	. . . . . . . . 9 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑎 ∩ 𝑏) = ((𝑢 ∩ 𝐴) ∩ (𝑣 ∩ 𝐴)))
33		inindir 4187	. . . . . . . . 9 ⊢ ((𝑢 ∩ 𝑣) ∩ 𝐴) = ((𝑢 ∩ 𝐴) ∩ (𝑣 ∩ 𝐴))
34	32, 33	eqtr4di 2786	. . . . . . . 8 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑎 ∩ 𝑏) = ((𝑢 ∩ 𝑣) ∩ 𝐴))
35	34	sseq2d 3964	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑤 ⊆ (𝑎 ∩ 𝑏) ↔ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
36	35	anbi2d 630	. . . . . . . . 9 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → ((𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏)) ↔ (𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))))
37	36	rexbidv 3158	. . . . . . . 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 3314	. . . . . . 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 247	. . . . . 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 3191	. . . . 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 240	. . . 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 3175	. . 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 7388	. . . 4 ⊢ (𝐵 ↾t 𝐴) ∈ V
44		isbasis2g 22873	. . . 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 234	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ TopBases)
47		relxp 5639	. . . . . 6 ⊢ Rel (V × V)
48		restfn 17338	. . . . . . . 8 ⊢ ↾t Fn (V × V)
49		fndm 6592	. . . . . . . 8 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V))
50	48, 49	ax-mp 5	. . . . . . 7 ⊢ dom ↾t = (V × V)
51	50	releqi 5724	. . . . . 6 ⊢ (Rel dom ↾t ↔ Rel (V × V))
52	47, 51	mpbir 231	. . . . 5 ⊢ Rel dom ↾t
53	52	ovprc2 7395	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵 ↾t 𝐴) = ∅)
54	53	adantl 481	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ ¬ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) = ∅)
55		fi0 9314	. . . 4 ⊢ (fi‘∅) = ∅
56		fibas 22902	. . . 4 ⊢ (fi‘∅) ∈ TopBases
57	55, 56	eqeltrri 2830	. . 3 ⊢ ∅ ∈ TopBases
58	54, 57	eqeltrdi 2841	. 2 ⊢ ((𝐵 ∈ TopBases ∧ ¬ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ TopBases)
59	46, 58	pm2.61dan 812	1 ⊢ (𝐵 ∈ TopBases → (𝐵 ↾t 𝐴) ∈ TopBases)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ∩ cin 3898   ⊆ wss 3899  ∅c0 4284   × cxp 5619  dom cdm 5621  Rel wrel 5626   Fn wfn 6484  ‘cfv 6489  (class class class)co 7355  ficfi 9304   ↾_t crest 17334  TopBasesctb 22870

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-en 8879  df-fin 8882  df-fi 9305  df-rest 17336  df-bases 22871

This theorem is referenced by:  resttop  23085  2ndcrest  23379