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Theorem cmetss 23021
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.)
Hypothesis
Ref Expression
cmetss.2 𝐽 = (MetOpen‘𝐷)
Assertion
Ref Expression
cmetss (𝐷 ∈ (CMet‘𝑋) → ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ 𝑌 ∈ (Clsd‘𝐽)))

Proof of Theorem cmetss
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmetmet 22992 . . . . . . . . 9 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2 metxmet 22049 . . . . . . . . 9 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
31, 2syl 17 . . . . . . . 8 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
43adantr 481 . . . . . . 7 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝐷 ∈ (∞Met‘𝑋))
5 cmetss.2 . . . . . . . 8 𝐽 = (MetOpen‘𝐷)
65mopntopon 22154 . . . . . . 7 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
74, 6syl 17 . . . . . 6 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
8 resss 5381 . . . . . . . 8 (𝐷 ↾ (𝑌 × 𝑌)) ⊆ 𝐷
9 dmss 5283 . . . . . . . 8 ((𝐷 ↾ (𝑌 × 𝑌)) ⊆ 𝐷 → dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom 𝐷)
10 dmss 5283 . . . . . . . 8 (dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom 𝐷 → dom dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom dom 𝐷)
118, 9, 10mp2b 10 . . . . . . 7 dom dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom dom 𝐷
12 cmetmet 22992 . . . . . . . . 9 ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
13 metdmdm 22051 . . . . . . . . 9 ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌) → 𝑌 = dom dom (𝐷 ↾ (𝑌 × 𝑌)))
1412, 13syl 17 . . . . . . . 8 ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) → 𝑌 = dom dom (𝐷 ↾ (𝑌 × 𝑌)))
15 metdmdm 22051 . . . . . . . . 9 (𝐷 ∈ (Met‘𝑋) → 𝑋 = dom dom 𝐷)
161, 15syl 17 . . . . . . . 8 (𝐷 ∈ (CMet‘𝑋) → 𝑋 = dom dom 𝐷)
17 sseq12 3607 . . . . . . . 8 ((𝑌 = dom dom (𝐷 ↾ (𝑌 × 𝑌)) ∧ 𝑋 = dom dom 𝐷) → (𝑌𝑋 ↔ dom dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom dom 𝐷))
1814, 16, 17syl2anr 495 . . . . . . 7 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (𝑌𝑋 ↔ dom dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom dom 𝐷))
1911, 18mpbiri 248 . . . . . 6 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌𝑋)
20 flimcls 21699 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝑌) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓))))
217, 19, 20syl2anc 692 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (𝑥 ∈ ((cls‘𝐽)‘𝑌) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓))))
22 simprrr 804 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝑓))
234adantr 481 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐷 ∈ (∞Met‘𝑋))
245methaus 22235 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus)
25 hausflimi 21694 . . . . . . . . 9 (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
2623, 24, 253syl 18 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
2723, 6syl 17 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐽 ∈ (TopOn‘𝑋))
28 simprl 793 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘𝑋))
29 simprrl 803 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑌𝑓)
30 flimrest 21697 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑌𝑓) → ((𝐽t 𝑌) fLim (𝑓t 𝑌)) = ((𝐽 fLim 𝑓) ∩ 𝑌))
3127, 28, 29, 30syl3anc 1323 . . . . . . . . . . 11 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → ((𝐽t 𝑌) fLim (𝑓t 𝑌)) = ((𝐽 fLim 𝑓) ∩ 𝑌))
3219adantr 481 . . . . . . . . . . . . 13 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑌𝑋)
33 eqid 2621 . . . . . . . . . . . . . 14 (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌))
34 eqid 2621 . . . . . . . . . . . . . 14 (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))
3533, 5, 34metrest 22239 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
3623, 32, 35syl2anc 692 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝐽t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
3736oveq1d 6619 . . . . . . . . . . 11 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → ((𝐽t 𝑌) fLim (𝑓t 𝑌)) = ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑓t 𝑌)))
3831, 37eqtr3d 2657 . . . . . . . . . 10 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → ((𝐽 fLim 𝑓) ∩ 𝑌) = ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑓t 𝑌)))
39 simplr 791 . . . . . . . . . . 11 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
405flimcfil 23020 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝑓 ∈ (CauFil‘𝐷))
4123, 22, 40syl2anc 692 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (CauFil‘𝐷))
42 cfilres 23002 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑌𝑓) → (𝑓 ∈ (CauFil‘𝐷) ↔ (𝑓t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))))
4323, 28, 29, 42syl3anc 1323 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝑓 ∈ (CauFil‘𝐷) ↔ (𝑓t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))))
4441, 43mpbid 222 . . . . . . . . . . 11 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝑓t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))
4534cmetcvg 22991 . . . . . . . . . . 11 (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ∧ (𝑓t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑓t 𝑌)) ≠ ∅)
4639, 44, 45syl2anc 692 . . . . . . . . . 10 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑓t 𝑌)) ≠ ∅)
4738, 46eqnetrd 2857 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → ((𝐽 fLim 𝑓) ∩ 𝑌) ≠ ∅)
48 n0 3907 . . . . . . . . . 10 (((𝐽 fLim 𝑓) ∩ 𝑌) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝐽 fLim 𝑓) ∩ 𝑌))
49 elin 3774 . . . . . . . . . . 11 (𝑥 ∈ ((𝐽 fLim 𝑓) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥𝑌))
5049exbii 1771 . . . . . . . . . 10 (∃𝑥 𝑥 ∈ ((𝐽 fLim 𝑓) ∩ 𝑌) ↔ ∃𝑥(𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥𝑌))
5148, 50bitri 264 . . . . . . . . 9 (((𝐽 fLim 𝑓) ∩ 𝑌) ≠ ∅ ↔ ∃𝑥(𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥𝑌))
5247, 51sylib 208 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → ∃𝑥(𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥𝑌))
53 mopick 2534 . . . . . . . 8 ((∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ∧ ∃𝑥(𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥𝑌)) → (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥𝑌))
5426, 52, 53syl2anc 692 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥𝑌))
5522, 54mpd 15 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥𝑌)
5655rexlimdvaa 3025 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (∃𝑓 ∈ (Fil‘𝑋)(𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)) → 𝑥𝑌))
5721, 56sylbid 230 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (𝑥 ∈ ((cls‘𝐽)‘𝑌) → 𝑥𝑌))
5857ssrdv 3589 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → ((cls‘𝐽)‘𝑌) ⊆ 𝑌)
595mopntop 22155 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
604, 59syl 17 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝐽 ∈ Top)
615mopnuni 22156 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
624, 61syl 17 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑋 = 𝐽)
6319, 62sseqtrd 3620 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌 𝐽)
64 eqid 2621 . . . . 5 𝐽 = 𝐽
6564iscld4 20779 . . . 4 ((𝐽 ∈ Top ∧ 𝑌 𝐽) → (𝑌 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑌) ⊆ 𝑌))
6660, 63, 65syl2anc 692 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (𝑌 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑌) ⊆ 𝑌))
6758, 66mpbird 247 . 2 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌 ∈ (Clsd‘𝐽))
681adantr 481 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐷 ∈ (Met‘𝑋))
6964cldss 20743 . . . . . 6 (𝑌 ∈ (Clsd‘𝐽) → 𝑌 𝐽)
7069adantl 482 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 𝐽)
7168, 2, 613syl 18 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
7270, 71sseqtr4d 3621 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌𝑋)
73 metres2 22078 . . . 4 ((𝐷 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
7468, 72, 73syl2anc 692 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
753ad2antrr 761 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐷 ∈ (∞Met‘𝑋))
7672adantr 481 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌𝑋)
7775, 76, 35syl2anc 692 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
7877eqcomd 2627 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (𝐽t 𝑌))
79 metxmet 22049 . . . . . . . . . . 11 ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
8074, 79syl 17 . . . . . . . . . 10 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
81 cfilfil 22973 . . . . . . . . . 10 (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (Fil‘𝑌))
8280, 81sylan 488 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (Fil‘𝑌))
83 elfvdm 6177 . . . . . . . . . 10 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
8483ad2antrr 761 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑋 ∈ dom CMet)
85 trfg 21605 . . . . . . . . 9 ((𝑓 ∈ (Fil‘𝑌) ∧ 𝑌𝑋𝑋 ∈ dom CMet) → ((𝑋filGen𝑓) ↾t 𝑌) = 𝑓)
8682, 76, 84, 85syl3anc 1323 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝑋filGen𝑓) ↾t 𝑌) = 𝑓)
8786eqcomd 2627 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 = ((𝑋filGen𝑓) ↾t 𝑌))
8878, 87oveq12d 6622 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) = ((𝐽t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌)))
8975, 6syl 17 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐽 ∈ (TopOn‘𝑋))
90 filfbas 21562 . . . . . . . . . 10 (𝑓 ∈ (Fil‘𝑌) → 𝑓 ∈ (fBas‘𝑌))
9182, 90syl 17 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (fBas‘𝑌))
92 filsspw 21565 . . . . . . . . . . 11 (𝑓 ∈ (Fil‘𝑌) → 𝑓 ⊆ 𝒫 𝑌)
9382, 92syl 17 . . . . . . . . . 10 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ⊆ 𝒫 𝑌)
94 sspwb 4878 . . . . . . . . . . 11 (𝑌𝑋 ↔ 𝒫 𝑌 ⊆ 𝒫 𝑋)
9576, 94sylib 208 . . . . . . . . . 10 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
9693, 95sstrd 3593 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ⊆ 𝒫 𝑋)
97 fbasweak 21579 . . . . . . . . 9 ((𝑓 ∈ (fBas‘𝑌) ∧ 𝑓 ⊆ 𝒫 𝑋𝑋 ∈ dom CMet) → 𝑓 ∈ (fBas‘𝑋))
9891, 96, 84, 97syl3anc 1323 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (fBas‘𝑋))
99 fgcl 21592 . . . . . . . 8 (𝑓 ∈ (fBas‘𝑋) → (𝑋filGen𝑓) ∈ (Fil‘𝑋))
10098, 99syl 17 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (Fil‘𝑋))
101 ssfg 21586 . . . . . . . . 9 (𝑓 ∈ (fBas‘𝑋) → 𝑓 ⊆ (𝑋filGen𝑓))
10298, 101syl 17 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ⊆ (𝑋filGen𝑓))
103 filtop 21569 . . . . . . . . 9 (𝑓 ∈ (Fil‘𝑌) → 𝑌𝑓)
10482, 103syl 17 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌𝑓)
105102, 104sseldd 3584 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌 ∈ (𝑋filGen𝑓))
106 flimrest 21697 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen𝑓) ∈ (Fil‘𝑋) ∧ 𝑌 ∈ (𝑋filGen𝑓)) → ((𝐽t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌)) = ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌))
10789, 100, 105, 106syl3anc 1323 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝐽t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌)) = ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌))
108 flimclsi 21692 . . . . . . . . 9 (𝑌 ∈ (𝑋filGen𝑓) → (𝐽 fLim (𝑋filGen𝑓)) ⊆ ((cls‘𝐽)‘𝑌))
109105, 108syl 17 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ⊆ ((cls‘𝐽)‘𝑌))
110 cldcls 20756 . . . . . . . . 9 (𝑌 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑌) = 𝑌)
111110ad2antlr 762 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((cls‘𝐽)‘𝑌) = 𝑌)
112109, 111sseqtrd 3620 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ⊆ 𝑌)
113 df-ss 3569 . . . . . . 7 ((𝐽 fLim (𝑋filGen𝑓)) ⊆ 𝑌 ↔ ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌) = (𝐽 fLim (𝑋filGen𝑓)))
114112, 113sylib 208 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌) = (𝐽 fLim (𝑋filGen𝑓)))
11588, 107, 1143eqtrd 2659 . . . . 5 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) = (𝐽 fLim (𝑋filGen𝑓)))
116 simpll 789 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐷 ∈ (CMet‘𝑋))
11768, 2syl 17 . . . . . . 7 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐷 ∈ (∞Met‘𝑋))
118 cfilresi 23001 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (CauFil‘𝐷))
119117, 118sylan 488 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (CauFil‘𝐷))
1205cmetcvg 22991 . . . . . 6 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑋filGen𝑓) ∈ (CauFil‘𝐷)) → (𝐽 fLim (𝑋filGen𝑓)) ≠ ∅)
121116, 119, 120syl2anc 692 . . . . 5 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ≠ ∅)
122115, 121eqnetrd 2857 . . . 4 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅)
123122ralrimiva 2960 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ∀𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅)
12434iscmet 22990 . . 3 ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌) ∧ ∀𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅))
12574, 123, 124sylanbrc 697 . 2 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
12667, 125impbida 876 1 (𝐷 ∈ (CMet‘𝑋) → ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ 𝑌 ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  ∃*wmo 2470  wne 2790  wral 2907  wrex 2908  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402   × cxp 5072  dom cdm 5074  cres 5076  cfv 5847  (class class class)co 6604  t crest 16002  ∞Metcxmt 19650  Metcme 19651  fBascfbas 19653  filGencfg 19654  MetOpencmopn 19655  Topctop 20617  TopOnctopon 20618  Clsdccld 20730  clsccl 20732  Hauscha 21022  Filcfil 21559   fLim cflim 21648  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-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-icc 12124  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-haus 21029  df-fil 21560  df-flim 21653  df-cfil 22961  df-cmet 22963
This theorem is referenced by:  recmet  23028  cmsss  23055  bnsscmcl  27573  rrnheibor  33268
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