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Theorem relcmpcmet 23023
Description: If 𝐷 is a metric space such that all the balls of some fixed size are relatively compact, then 𝐷 is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
relcmpcmet.1 𝐽 = (MetOpen‘𝐷)
relcmpcmet.2 (𝜑𝐷 ∈ (Met‘𝑋))
relcmpcmet.3 (𝜑𝑅 ∈ ℝ+)
relcmpcmet.4 ((𝜑𝑥𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
Assertion
Ref Expression
relcmpcmet (𝜑𝐷 ∈ (CMet‘𝑋))
Distinct variable groups:   𝑥,𝐷   𝑥,𝐽   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋

Proof of Theorem relcmpcmet
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 relcmpcmet.2 . 2 (𝜑𝐷 ∈ (Met‘𝑋))
2 metxmet 22049 . . . . . . 7 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
31, 2syl 17 . . . . . 6 (𝜑𝐷 ∈ (∞Met‘𝑋))
43adantr 481 . . . . 5 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋))
5 simpr 477 . . . . 5 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (CauFil‘𝐷))
6 relcmpcmet.3 . . . . . 6 (𝜑𝑅 ∈ ℝ+)
76adantr 481 . . . . 5 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝑅 ∈ ℝ+)
8 cfil3i 22975 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
94, 5, 7, 8syl3anc 1323 . . . 4 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → ∃𝑥𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
103ad2antrr 761 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐷 ∈ (∞Met‘𝑋))
11 relcmpcmet.1 . . . . . . . . 9 𝐽 = (MetOpen‘𝐷)
1211mopntopon 22154 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
1310, 12syl 17 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
14 cfilfil 22973 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
153, 14sylan 488 . . . . . . . 8 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
1615adantr 481 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
17 simprr 795 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
18 topontop 20641 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1913, 18syl 17 . . . . . . . . . 10 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ Top)
20 simprl 793 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑥𝑋)
216rpxrd 11817 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ ℝ*)
2221ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑅 ∈ ℝ*)
23 blssm 22133 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑅 ∈ ℝ*) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
2410, 20, 22, 23syl3anc 1323 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
25 toponuni 20642 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2613, 25syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑋 = 𝐽)
2724, 26sseqtrd 3620 . . . . . . . . . 10 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝐽)
28 eqid 2621 . . . . . . . . . . 11 𝐽 = 𝐽
2928clsss3 20773 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ 𝐽) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝐽)
3019, 27, 29syl2anc 692 . . . . . . . . 9 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝐽)
3130, 26sseqtr4d 3621 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋)
3228sscls 20770 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ 𝐽) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
3319, 27, 32syl2anc 692 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
34 filss 21567 . . . . . . . 8 ((𝑓 ∈ (Fil‘𝑋) ∧ ((𝑥(ball‘𝐷)𝑅) ∈ 𝑓 ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
3516, 17, 31, 33, 34syl13anc 1325 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
36 fclsrest 21738 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
3713, 16, 35, 36syl3anc 1323 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
38 inss1 3811 . . . . . . 7 ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fClus 𝑓)
39 eqid 2621 . . . . . . . . 9 dom dom 𝐷 = dom dom 𝐷
4011, 39cfilfcls 22980 . . . . . . . 8 (𝑓 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
4140ad2antlr 762 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
4238, 41syl5sseq 3632 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fLim 𝑓))
4337, 42eqsstrd 3618 . . . . 5 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓))
44 relcmpcmet.4 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
4544ad2ant2r 782 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
46 filfbas 21562 . . . . . . . . . 10 (𝑓 ∈ (Fil‘𝑋) → 𝑓 ∈ (fBas‘𝑋))
4716, 46syl 17 . . . . . . . . 9 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (fBas‘𝑋))
48 fbncp 21553 . . . . . . . . 9 ((𝑓 ∈ (fBas‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
4947, 35, 48syl2anc 692 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
50 trfil3 21602 . . . . . . . . 9 ((𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → ((𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
5116, 31, 50syl2anc 692 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
5249, 51mpbird 247 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
53 resttopon 20875 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
5413, 31, 53syl2anc 692 . . . . . . . . 9 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
55 toponuni 20642 . . . . . . . . 9 ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
5654, 55syl 17 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
5756fveq2d 6152 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (Fil‘ (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
5852, 57eleqtrd 2700 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘ (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
59 eqid 2621 . . . . . . 7 (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
6059fclscmpi 21743 . . . . . 6 (((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp ∧ (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘ (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
6145, 58, 60syl2anc 692 . . . . 5 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
62 ssn0 3948 . . . . 5 ((((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓) ∧ ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅) → (𝐽 fLim 𝑓) ≠ ∅)
6343, 61, 62syl2anc 692 . . . 4 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fLim 𝑓) ≠ ∅)
649, 63rexlimddv 3028 . . 3 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑓) ≠ ∅)
6564ralrimiva 2960 . 2 (𝜑 → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)
6611iscmet 22990 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
671, 65, 66sylanbrc 697 1 (𝜑𝐷 ∈ (CMet‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  cdif 3552  cin 3554  wss 3555  c0 3891   cuni 4402  dom cdm 5074  cfv 5847  (class class class)co 6604  *cxr 10017  +crp 11776  t crest 16002  ∞Metcxmt 19650  Metcme 19651  ballcbl 19652  fBascfbas 19653  MetOpencmopn 19655  Topctop 20617  TopOnctopon 20618  clsccl 20732  Compccmp 21099  Filcfil 21559   fLim cflim 21648   fClus cfcls 21650  CauFilccfil 22958  CMetcms 22960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ico 12123  df-rest 16004  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-fbas 19662  df-fg 19663  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-cmp 21100  df-fil 21560  df-flim 21653  df-fcls 21655  df-cfil 22961  df-cmet 22963
This theorem is referenced by:  cmpcmet  23024  cncmet  23027
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