Theorem restbas 23106

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 17351	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝑎 ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐵 𝑎 = (𝑢 ∩ 𝐴)))
2		elrest 17351	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝑏 ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑣 ∈ 𝐵 𝑏 = (𝑣 ∩ 𝐴)))
3	1, 2	anbi12d 633	. . . . . 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 3209	. . . . . 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 775	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝐵 ∈ TopBases)
7		simplrl 777	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑢 ∈ 𝐵)
8		simplrr 778	. . . . . . . . . 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 4157	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑐 ∈ (𝑢 ∩ 𝑣))
11		basis2 22899	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ∈ 𝐵) ∧ (𝑣 ∈ 𝐵 ∧ 𝑐 ∈ (𝑢 ∩ 𝑣))) → ∃𝑧 ∈ 𝐵 (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))
12	6, 7, 8, 10, 11	syl22anc 839	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → ∃𝑧 ∈ 𝐵 (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))
13		simplll 775	. . . . . . . . . . . 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 771	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑧 ∈ 𝐵)
17		elrestr 17352	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (𝑧 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
18	14, 15, 16, 17	syl3anc 1374	. . . . . . . . . 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 781	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑐 ∈ 𝑧)
20		simplr 769	. . . . . . . . . . . 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 4158	. . . . . . . . . . 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 4153	. . . . . . . . . 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 782	. . . . . . . . . . 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 4197	. . . . . . . . . 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 2826	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → (𝑐 ∈ 𝑤 ↔ 𝑐 ∈ (𝑧 ∩ 𝐴)))
26		sseq1 3960	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → (𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴) ↔ (𝑧 ∩ 𝐴) ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
27	25, 26	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → ((𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ↔ (𝑐 ∈ (𝑧 ∩ 𝐴) ∧ (𝑧 ∩ 𝐴) ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))))
28	27	rspcev 3577	. . . . . . . . . 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 837	. . . . . . . . 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 3144	. . . . . . . 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 3129	. . . . . . 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 4168	. . . . . . . . 9 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑎 ∩ 𝑏) = ((𝑢 ∩ 𝐴) ∩ (𝑣 ∩ 𝐴)))
33		inindir 4189	. . . . . . . . 9 ⊢ ((𝑢 ∩ 𝑣) ∩ 𝐴) = ((𝑢 ∩ 𝐴) ∩ (𝑣 ∩ 𝐴))
34	32, 33	eqtr4di 2790	. . . . . . . 8 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑎 ∩ 𝑏) = ((𝑢 ∩ 𝑣) ∩ 𝐴))
35	34	sseq2d 3967	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑤 ⊆ (𝑎 ∩ 𝑏) ↔ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
36	35	anbi2d 631	. . . . . . . . 9 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → ((𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏)) ↔ (𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))))
37	36	rexbidv 3161	. . . . . . . 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 3317	. . . . . . 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 3194	. . . . 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 3178	. . 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 7393	. . . 4 ⊢ (𝐵 ↾t 𝐴) ∈ V
44		isbasis2g 22896	. . . 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 5643	. . . . . 6 ⊢ Rel (V × V)
48		restfn 17348	. . . . . . . 8 ⊢ ↾t Fn (V × V)
49		fndm 6596	. . . . . . . 8 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V))
50	48, 49	ax-mp 5	. . . . . . 7 ⊢ dom ↾t = (V × V)
51	50	releqi 5728	. . . . . 6 ⊢ (Rel dom ↾t ↔ Rel (V × V))
52	47, 51	mpbir 231	. . . . 5 ⊢ Rel dom ↾t
53	52	ovprc2 7400	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵 ↾t 𝐴) = ∅)
54	53	adantl 481	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ ¬ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) = ∅)
55		fi0 9327	. . . 4 ⊢ (fi‘∅) = ∅
56		fibas 22925	. . . 4 ⊢ (fi‘∅) ∈ TopBases
57	55, 56	eqeltrri 2834	. . 3 ⊢ ∅ ∈ TopBases
58	54, 57	eqeltrdi 2845	. 2 ⊢ ((𝐵 ∈ TopBases ∧ ¬ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ TopBases)
59	46, 58	pm2.61dan 813	1 ⊢ (𝐵 ∈ TopBases → (𝐵 ↾t 𝐴) ∈ TopBases)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286   × cxp 5623  dom cdm 5625  Rel wrel 5630   Fn wfn 6488  ‘cfv 6493  (class class class)co 7360  ficfi 9317   ↾_t crest 17344  TopBasesctb 22893

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-en 8888  df-fin 8891  df-fi 9318  df-rest 17346  df-bases 22894

This theorem is referenced by:  resttop  23108  2ndcrest  23402