Theorem cmetss 24213

Description: A subspace of a complete metric space is complete iff it is closed in the parent space. Theorem 1.4-7 of [Kreyszig] p. 30. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 15-Oct-2015.) (Proof shortened by AV, 9-Oct-2022.)

Hypothesis

Ref	Expression
metsscmetcld.j	⊢ 𝐽 = (MetOpen‘𝐷)

Assertion

Ref	Expression
cmetss	⊢ (𝐷 ∈ (CMet‘𝑋) → ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ 𝑌 ∈ (Clsd‘𝐽)))

Proof of Theorem cmetss

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmetmet 24183	. . 3 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2		metsscmetcld.j	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐷)
3	2	metsscmetcld 24212	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌 ∈ (Clsd‘𝐽))
4	1, 3	sylan 583	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌 ∈ (Clsd‘𝐽))
5	1	adantr 484	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐷 ∈ (Met‘𝑋))
6		eqid 2737	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	cldss 21926	. . . . . 6 ⊢ (𝑌 ∈ (Clsd‘𝐽) → 𝑌 ⊆ ∪ 𝐽)
8	7	adantl 485	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ ∪ 𝐽)
9		metxmet 23232	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
10	2	mopnuni 23339	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
11	5, 9, 10	3syl 18	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑋 = ∪ 𝐽)
12	8, 11	sseqtrrd 3942	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ 𝑋)
13		metres2 23261	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
14	5, 12, 13	syl2anc 587	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
15	1, 9	syl 17	. . . . . . . . . 10 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
16	15	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐷 ∈ (∞Met‘𝑋))
17	12	adantr 484	. . . . . . . . 9 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌 ⊆ 𝑋)
18		eqid 2737	. . . . . . . . . 10 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌))
19		eqid 2737	. . . . . . . . . 10 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))
20	18, 2, 19	metrest 23422	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
21	16, 17, 20	syl2anc 587	. . . . . . . 8 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
22	21	eqcomd 2743	. . . . . . 7 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (𝐽 ↾t 𝑌))
23		metxmet 23232	. . . . . . . . . . 11 ⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
24	14, 23	syl 17	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
25		cfilfil 24164	. . . . . . . . . 10 ⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (Fil‘𝑌))
26	24, 25	sylan 583	. . . . . . . . 9 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (Fil‘𝑌))
27		elfvdm 6749	. . . . . . . . . 10 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
28	27	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑋 ∈ dom CMet)
29		trfg 22788	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Fil‘𝑌) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ dom CMet) → ((𝑋filGen𝑓) ↾t 𝑌) = 𝑓)
30	26, 17, 28, 29	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝑋filGen𝑓) ↾t 𝑌) = 𝑓)
31	30	eqcomd 2743	. . . . . . 7 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 = ((𝑋filGen𝑓) ↾t 𝑌))
32	22, 31	oveq12d 7231	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) = ((𝐽 ↾t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌)))
33	2	mopntopon 23337	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34	16, 33	syl 17	. . . . . . 7 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐽 ∈ (TopOn‘𝑋))
35		filfbas 22745	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ∈ (fBas‘𝑌))
36	26, 35	syl 17	. . . . . . . . 9 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (fBas‘𝑌))
37		filsspw 22748	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ⊆ 𝒫 𝑌)
38	26, 37	syl 17	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ⊆ 𝒫 𝑌)
39	17	sspwd 4528	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
40	38, 39	sstrd 3911	. . . . . . . . 9 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ⊆ 𝒫 𝑋)
41		fbasweak 22762	. . . . . . . . 9 ⊢ ((𝑓 ∈ (fBas‘𝑌) ∧ 𝑓 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ dom CMet) → 𝑓 ∈ (fBas‘𝑋))
42	36, 40, 28, 41	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (fBas‘𝑋))
43		fgcl 22775	. . . . . . . 8 ⊢ (𝑓 ∈ (fBas‘𝑋) → (𝑋filGen𝑓) ∈ (Fil‘𝑋))
44	42, 43	syl 17	. . . . . . 7 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (Fil‘𝑋))
45		ssfg 22769	. . . . . . . . 9 ⊢ (𝑓 ∈ (fBas‘𝑋) → 𝑓 ⊆ (𝑋filGen𝑓))
46	42, 45	syl 17	. . . . . . . 8 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ⊆ (𝑋filGen𝑓))
47		filtop 22752	. . . . . . . . 9 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝑓)
48	26, 47	syl 17	. . . . . . . 8 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌 ∈ 𝑓)
49	46, 48	sseldd 3902	. . . . . . 7 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌 ∈ (𝑋filGen𝑓))
50		flimrest 22880	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen𝑓) ∈ (Fil‘𝑋) ∧ 𝑌 ∈ (𝑋filGen𝑓)) → ((𝐽 ↾t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌)) = ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌))
51	34, 44, 49, 50	syl3anc 1373	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝐽 ↾t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌)) = ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌))
52		flimclsi 22875	. . . . . . . . 9 ⊢ (𝑌 ∈ (𝑋filGen𝑓) → (𝐽 fLim (𝑋filGen𝑓)) ⊆ ((cls‘𝐽)‘𝑌))
53	49, 52	syl 17	. . . . . . . 8 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ⊆ ((cls‘𝐽)‘𝑌))
54		cldcls 21939	. . . . . . . . 9 ⊢ (𝑌 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑌) = 𝑌)
55	54	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((cls‘𝐽)‘𝑌) = 𝑌)
56	53, 55	sseqtrd 3941	. . . . . . 7 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ⊆ 𝑌)
57		df-ss 3883	. . . . . . 7 ⊢ ((𝐽 fLim (𝑋filGen𝑓)) ⊆ 𝑌 ↔ ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌) = (𝐽 fLim (𝑋filGen𝑓)))
58	56, 57	sylib 221	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌) = (𝐽 fLim (𝑋filGen𝑓)))
59	32, 51, 58	3eqtrd 2781	. . . . 5 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) = (𝐽 fLim (𝑋filGen𝑓)))
60		simpll 767	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐷 ∈ (CMet‘𝑋))
61	5, 9	syl 17	. . . . . . 7 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐷 ∈ (∞Met‘𝑋))
62		cfilresi 24192	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (CauFil‘𝐷))
63	61, 62	sylan 583	. . . . . 6 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (CauFil‘𝐷))
64	2	cmetcvg 24182	. . . . . 6 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑋filGen𝑓) ∈ (CauFil‘𝐷)) → (𝐽 fLim (𝑋filGen𝑓)) ≠ ∅)
65	60, 63, 64	syl2anc 587	. . . . 5 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ≠ ∅)
66	59, 65	eqnetrd 3008	. . . 4 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅)
67	66	ralrimiva 3105	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ∀𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅)
68	19	iscmet 24181	. . 3 ⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌) ∧ ∀𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅))
69	14, 67, 68	sylanbrc 586	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
70	4, 69	impbida 801	1 ⊢ (𝐷 ∈ (CMet‘𝑋) → ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ 𝑌 ∈ (Clsd‘𝐽)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  𝒫 cpw 4513  ∪ cuni 4819   × cxp 5549  dom cdm 5551   ↾ cres 5553  ‘cfv 6380  (class class class)co 7213   ↾_t crest 16925  ∞Metcxmet 20348  Metcmet 20349  fBascfbas 20351  filGencfg 20352  MetOpencmopn 20353  TopOnctopon 21807  Clsdccld 21913  clsccl 21915  Filcfil 22742   fLim cflim 22831  CauFilccfil 24149  CMetccmet 24151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fi 9027  df-sup 9058  df-inf 9059  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-n0 12091  df-z 12177  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-ico 12941  df-icc 12942  df-rest 16927  df-topgen 16948  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-mopn 20359  df-fbas 20360  df-fg 20361  df-top 21791  df-topon 21808  df-bases 21843  df-cld 21916  df-ntr 21917  df-cls 21918  df-nei 21995  df-haus 22212  df-fil 22743  df-flim 22836  df-cfil 24152  df-cmet 24154

This theorem is referenced by:  recmet  24220  cmsss  24248  cmscsscms  24270  bnsscmcl  28949  rrnheibor  35732