Theorem restbas 21766

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 16701	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝑎 ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐵 𝑎 = (𝑢 ∩ 𝐴)))
2		elrest 16701	. . . . . . 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 3367	. . . . . 6 ⊢ (∃𝑢 ∈ 𝐵 ∃𝑣 ∈ 𝐵 (𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) ↔ (∃𝑢 ∈ 𝐵 𝑎 = (𝑢 ∩ 𝐴) ∧ ∃𝑣 ∈ 𝐵 𝑏 = (𝑣 ∩ 𝐴)))
5	3, 4	syl6bbr 291	. . . . 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 773	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝐵 ∈ TopBases)
7		simplrl 775	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑢 ∈ 𝐵)
8		simplrr 776	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑣 ∈ 𝐵)
9		simpr 487	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴))
10	9	elin1d 4175	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) → 𝑐 ∈ (𝑢 ∩ 𝑣))
11		basis2 21559	. . . . . . . . . 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 773	. . . . . . . . . . . 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 497	. . . . . . . . . . 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 498	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝐴 ∈ V)
16		simprl 769	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑧 ∈ 𝐵)
17		elrestr 16702	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (𝑧 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
18	14, 15, 16, 17	syl3anc 1367	. . . . . . . . . 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 779	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑐 ∈ ((𝑢 ∩ 𝑣) ∩ 𝐴)) ∧ (𝑧 ∈ 𝐵 ∧ (𝑐 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑢 ∩ 𝑣)))) → 𝑐 ∈ 𝑧)
20		simplr 767	. . . . . . . . . . . 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 4176	. . . . . . . . . . 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 4171	. . . . . . . . . 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 780	. . . . . . . . . . 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 4212	. . . . . . . . . 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 2901	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑧 ∩ 𝐴) → (𝑐 ∈ 𝑤 ↔ 𝑐 ∈ (𝑧 ∩ 𝐴)))
26		sseq1 3992	. . . . . . . . . . . 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 3623	. . . . . . . . . 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 3291	. . . . . . . 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 3182	. . . . . . 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 4184	. . . . . . . . 9 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑎 ∩ 𝑏) = ((𝑢 ∩ 𝐴) ∩ (𝑣 ∩ 𝐴)))
33		inindir 4204	. . . . . . . . 9 ⊢ ((𝑢 ∩ 𝑣) ∩ 𝐴) = ((𝑢 ∩ 𝐴) ∩ (𝑣 ∩ 𝐴))
34	32, 33	syl6eqr 2874	. . . . . . . 8 ⊢ ((𝑎 = (𝑢 ∩ 𝐴) ∧ 𝑏 = (𝑣 ∩ 𝐴)) → (𝑎 ∩ 𝑏) = ((𝑢 ∩ 𝑣) ∩ 𝐴))
35	34	sseq2d 3999	. . . . . . . . . 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 3297	. . . . . . . 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 3401	. . . . . . 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 249	. . . . . 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 3294	. . . . 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 242	. . . 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 3190	. . 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 7189	. . . 4 ⊢ (𝐵 ↾t 𝐴) ∈ V
44		isbasis2g 21556	. . . 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 236	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ TopBases)
47		relxp 5573	. . . . . 6 ⊢ Rel (V × V)
48		restfn 16698	. . . . . . . 8 ⊢ ↾t Fn (V × V)
49		fndm 6455	. . . . . . . 8 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V))
50	48, 49	ax-mp 5	. . . . . . 7 ⊢ dom ↾t = (V × V)
51	50	releqi 5652	. . . . . 6 ⊢ (Rel dom ↾t ↔ Rel (V × V))
52	47, 51	mpbir 233	. . . . 5 ⊢ Rel dom ↾t
53	52	ovprc2 7196	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵 ↾t 𝐴) = ∅)
54	53	adantl 484	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ ¬ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) = ∅)
55		fi0 8884	. . . 4 ⊢ (fi‘∅) = ∅
56		fibas 21585	. . . 4 ⊢ (fi‘∅) ∈ TopBases
57	55, 56	eqeltrri 2910	. . 3 ⊢ ∅ ∈ TopBases
58	54, 57	eqeltrdi 2921	. 2 ⊢ ((𝐵 ∈ TopBases ∧ ¬ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ TopBases)
59	46, 58	pm2.61dan 811	1 ⊢ (𝐵 ∈ TopBases → (𝐵 ↾t 𝐴) ∈ TopBases)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291   × cxp 5553  dom cdm 5555  Rel wrel 5560   Fn wfn 6350  ‘cfv 6355  (class class class)co 7156  ficfi 8874   ↾_t crest 16694  TopBasesctb 21553

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-oadd 8106  df-er 8289  df-en 8510  df-fin 8513  df-fi 8875  df-rest 16696  df-bases 21554

This theorem is referenced by:  resttop  21768  2ndcrest  22062