Theorem restbas 23506

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 17627	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝑎 ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐵 𝑎 = (𝑢 ∩ 𝐴)))
2		elrest 17627	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝑏 ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑣 ∈ 𝐵 𝑏 = (𝑣 ∩ 𝐴)))
3	1, 2	anbi12d 644	. . . . . 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 3238	. . . . . 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 292	. . . . 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 787	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝐵 ∈ TopBases)
7		simplrl 789	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑢 ∈ 𝐵)
8		simplrr 790	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑣 ∈ 𝐵)
9		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴))
10	9	elin1d 4104	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑐 ∈ (𝑢 ∩ 𝑣))
11		basis2 23299	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ∈ 𝐵) ∧ (𝑣 ∈ 𝐵 ∧ 𝑐 ∈ (𝑢 ∩ 𝑣))) → ∃𝑧 ∈ 𝐵 (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))
12	6, 7, 8, 10, 11	syl22anc 852	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → ∃𝑧 ∈ 𝐵 (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))
13		simplll 787	. . . . . . . . . . . 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 500	. . . . . . . . . . 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 501	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝐴 ∈ V)
16		simprl 783	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑧 ∈ 𝐵)
17		elrestr 17628	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (𝑧 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
18	14, 15, 16, 17	syl3anc 1399	. . . . . . . . . 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 793	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑐 ∈ 𝑧)
20		simplr 781	. . . . . . . . . . . 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 4105	. . . . . . . . . . 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 4100	. . . . . . . . . 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 794	. . . . . . . . . . 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 4143	. . . . . . . . . 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 2853	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → (𝑐 ∈ 𝑤 ↔ 𝑐 ∈ (𝑧 ∩ 𝐴)))
26		sseq1 3938	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → (𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴) ↔ (𝑧 ∩ 𝐴) ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
27	25, 26	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → ((𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ↔ (𝑐 ∈ (𝑧 ∩ 𝐴) ∧ (𝑧 ∩ 𝐴) ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))))
28	27	rspcev 3580	. . . . . . . . . 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 850	. . . . . . . . 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 3173	. . . . . . . 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 3158	. . . . . . 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 4115	. . . . . . . . 9 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑎 ∩ 𝑏) = ((𝑢 ∩ 𝐴) ∩ (𝑣 ∩ 𝐴)))
33		inindir 4135	. . . . . . . . 9 ⊢ ((𝑢 ∩ 𝑣) ∩ 𝐴) = ((𝑢 ∩ 𝐴) ∩ (𝑣 ∩ 𝐴))
34	32, 33	eqtr4di 2816	. . . . . . . 8 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑎 ∩ 𝑏) = ((𝑢 ∩ 𝑣) ∩ 𝐴))
35	34	sseq2d 3945	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑤 ⊆ (𝑎 ∩ 𝑏) ↔ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴)))
36	35	anbi2d 642	. . . . . . . . 9 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → ((𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝑏)) ↔ (𝑐 ∈ 𝑤 ∧ 𝑤 ⊆ ((𝑢 ∩ 𝑣) ∩ 𝐴))))
37	36	rexbidv 3190	. . . . . . . 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 3338	. . . . . . 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 250	. . . . . 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 243	. . . 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 3207	. . 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 7476	. . . 4 ⊢ (𝐵 ↾t 𝐴) ∈ V
44		isbasis2g 23296	. . . 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 237	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ TopBases)
47		relxp 5673	. . . . . 6 ⊢ Rel (V × V)
48		restfn 17624	. . . . . . . 8 ⊢ ↾t Fn (V × V)
49		fndm 6664	. . . . . . . 8 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V))
50	48, 49	ax-mp 5	. . . . . . 7 ⊢ dom ↾t = (V × V)
51	50	releqi 5759	. . . . . 6 ⊢ (Rel dom ↾t ↔ Rel (V × V))
52	47, 51	mpbir 234	. . . . 5 ⊢ Rel dom ↾t
53	52	ovprc2 7483	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵 ↾t 𝐴) = ∅)
54	53	adantl 487	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ ¬ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) = ∅)
55		fi0 9439	. . . 4 ⊢ (fi‘∅) = ∅
56		fibas 23325	. . . 4 ⊢ (fi‘∅) ∈ TopBases
57	55, 56	eqeltrri 2861	. . 3 ⊢ ∅ ∈ TopBases
58	54, 57	eqeltrdi 2872	. 2 ⊢ ((𝐵 ∈ TopBases ∧ ¬ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ TopBases)
59	46, 58	pm2.61dan 825	1 ⊢ (𝐵 ∈ TopBases → (𝐵 ↾t 𝐴) ∈ TopBases)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281   × cxp 5653  dom cdm 5655  Rel wrel 5660   Fn wfn 6553  ‘cfv 6558  (class class class)co 7443  ficfi 9429   ↾_t crest 17620  TopBasesctb 23293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-en 9002  df-fin 9005  df-fi 9430  df-rest 17622  df-bases 23294

This theorem is used by:  resttop  23508  2ndcrest  23802