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Theorem cmetss 23159
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 23130 . . . . . . . . 9 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2 metxmet 22186 . . . . . . . . 9 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
31, 2syl 17 . . . . . . . 8 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
43adantr 480 . . . . . . 7 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝐷 ∈ (∞Met‘𝑋))
5 cmetss.2 . . . . . . . 8 𝐽 = (MetOpen‘𝐷)
65mopntopon 22291 . . . . . . 7 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
74, 6syl 17 . . . . . 6 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
8 resss 5457 . . . . . . . 8 (𝐷 ↾ (𝑌 × 𝑌)) ⊆ 𝐷
9 dmss 5355 . . . . . . . 8 ((𝐷 ↾ (𝑌 × 𝑌)) ⊆ 𝐷 → dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom 𝐷)
10 dmss 5355 . . . . . . . 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 23130 . . . . . . . . 9 ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
13 metdmdm 22188 . . . . . . . . 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 22188 . . . . . . . . 9 (𝐷 ∈ (Met‘𝑋) → 𝑋 = dom dom 𝐷)
161, 15syl 17 . . . . . . . 8 (𝐷 ∈ (CMet‘𝑋) → 𝑋 = dom dom 𝐷)
17 sseq12 3661 . . . . . . . 8 ((𝑌 = dom dom (𝐷 ↾ (𝑌 × 𝑌)) ∧ 𝑋 = dom dom 𝐷) → (𝑌𝑋 ↔ dom dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom dom 𝐷))
1814, 16, 17syl2anr 494 . . . . . . 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 21836 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝑌) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓))))
217, 19, 20syl2anc 694 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (𝑥 ∈ ((cls‘𝐽)‘𝑌) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓))))
22 simprrr 822 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝑓))
234adantr 480 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐷 ∈ (∞Met‘𝑋))
245methaus 22372 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus)
25 hausflimi 21831 . . . . . . . . 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 809 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘𝑋))
29 simprrl 821 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑌𝑓)
30 flimrest 21834 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑌𝑓) → ((𝐽t 𝑌) fLim (𝑓t 𝑌)) = ((𝐽 fLim 𝑓) ∩ 𝑌))
3127, 28, 29, 30syl3anc 1366 . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑌𝑋)
33 eqid 2651 . . . . . . . . . . . . . 14 (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌))
34 eqid 2651 . . . . . . . . . . . . . 14 (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))
3533, 5, 34metrest 22376 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
3623, 32, 35syl2anc 694 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝐽t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
3736oveq1d 6705 . . . . . . . . . . 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 2687 . . . . . . . . . 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 807 . . . . . . . . . . 11 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
405flimcfil 23158 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝑓 ∈ (CauFil‘𝐷))
4123, 22, 40syl2anc 694 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (CauFil‘𝐷))
42 cfilres 23140 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑌𝑓) → (𝑓 ∈ (CauFil‘𝐷) ↔ (𝑓t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))))
4323, 28, 29, 42syl3anc 1366 . . . . . . . . . . . 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 23129 . . . . . . . . . . 11 (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ∧ (𝑓t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑓t 𝑌)) ≠ ∅)
4639, 44, 45syl2anc 694 . . . . . . . . . 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 2890 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → ((𝐽 fLim 𝑓) ∩ 𝑌) ≠ ∅)
48 n0 3964 . . . . . . . . . 10 (((𝐽 fLim 𝑓) ∩ 𝑌) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝐽 fLim 𝑓) ∩ 𝑌))
49 elin 3829 . . . . . . . . . . 11 (𝑥 ∈ ((𝐽 fLim 𝑓) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥𝑌))
5049exbii 1814 . . . . . . . . . 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 2564 . . . . . . . 8 ((∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ∧ ∃𝑥(𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥𝑌)) → (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥𝑌))
5426, 52, 53syl2anc 694 . . . . . . 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 3061 . . . . 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 3642 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → ((cls‘𝐽)‘𝑌) ⊆ 𝑌)
595mopntop 22292 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
604, 59syl 17 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝐽 ∈ Top)
615mopnuni 22293 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
624, 61syl 17 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑋 = 𝐽)
6319, 62sseqtrd 3674 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌 𝐽)
64 eqid 2651 . . . . 5 𝐽 = 𝐽
6564iscld4 20917 . . . 4 ((𝐽 ∈ Top ∧ 𝑌 𝐽) → (𝑌 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑌) ⊆ 𝑌))
6660, 63, 65syl2anc 694 . . 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 480 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐷 ∈ (Met‘𝑋))
6964cldss 20881 . . . . . 6 (𝑌 ∈ (Clsd‘𝐽) → 𝑌 𝐽)
7069adantl 481 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 𝐽)
7168, 2, 613syl 18 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
7270, 71sseqtr4d 3675 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌𝑋)
73 metres2 22215 . . . 4 ((𝐷 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
7468, 72, 73syl2anc 694 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
753ad2antrr 762 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐷 ∈ (∞Met‘𝑋))
7672adantr 480 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌𝑋)
7775, 76, 35syl2anc 694 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
7877eqcomd 2657 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (𝐽t 𝑌))
79 metxmet 22186 . . . . . . . . . . 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 23111 . . . . . . . . . 10 (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (Fil‘𝑌))
8280, 81sylan 487 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (Fil‘𝑌))
83 elfvdm 6258 . . . . . . . . . 10 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
8483ad2antrr 762 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑋 ∈ dom CMet)
85 trfg 21742 . . . . . . . . 9 ((𝑓 ∈ (Fil‘𝑌) ∧ 𝑌𝑋𝑋 ∈ dom CMet) → ((𝑋filGen𝑓) ↾t 𝑌) = 𝑓)
8682, 76, 84, 85syl3anc 1366 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝑋filGen𝑓) ↾t 𝑌) = 𝑓)
8786eqcomd 2657 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 = ((𝑋filGen𝑓) ↾t 𝑌))
8878, 87oveq12d 6708 . . . . . 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 21699 . . . . . . . . . 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 21702 . . . . . . . . . . 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 4947 . . . . . . . . . . 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 3646 . . . . . . . . 9 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ⊆ 𝒫 𝑋)
97 fbasweak 21716 . . . . . . . . 9 ((𝑓 ∈ (fBas‘𝑌) ∧ 𝑓 ⊆ 𝒫 𝑋𝑋 ∈ dom CMet) → 𝑓 ∈ (fBas‘𝑋))
9891, 96, 84, 97syl3anc 1366 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (fBas‘𝑋))
99 fgcl 21729 . . . . . . . 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 21723 . . . . . . . . 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 21706 . . . . . . . . 9 (𝑓 ∈ (Fil‘𝑌) → 𝑌𝑓)
10482, 103syl 17 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌𝑓)
105102, 104sseldd 3637 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌 ∈ (𝑋filGen𝑓))
106 flimrest 21834 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen𝑓) ∈ (Fil‘𝑋) ∧ 𝑌 ∈ (𝑋filGen𝑓)) → ((𝐽t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌)) = ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌))
10789, 100, 105, 106syl3anc 1366 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝐽t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌)) = ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌))
108 flimclsi 21829 . . . . . . . . 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 20894 . . . . . . . . 9 (𝑌 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑌) = 𝑌)
111110ad2antlr 763 . . . . . . . 8 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((cls‘𝐽)‘𝑌) = 𝑌)
112109, 111sseqtrd 3674 . . . . . . 7 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ⊆ 𝑌)
113 df-ss 3621 . . . . . . 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 2689 . . . . 5 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) = (𝐽 fLim (𝑋filGen𝑓)))
116 simpll 805 . . . . . 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 23139 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (CauFil‘𝐷))
119117, 118sylan 487 . . . . . 6 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (CauFil‘𝐷))
1205cmetcvg 23129 . . . . . 6 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑋filGen𝑓) ∈ (CauFil‘𝐷)) → (𝐽 fLim (𝑋filGen𝑓)) ≠ ∅)
121116, 119, 120syl2anc 694 . . . . 5 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ≠ ∅)
122115, 121eqnetrd 2890 . . . 4 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅)
123122ralrimiva 2995 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ∀𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅)
12434iscmet 23128 . . 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 699 . 2 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
12667, 125impbida 895 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 383   = wceq 1523  wex 1744  wcel 2030  ∃*wmo 2499  wne 2823  wral 2941  wrex 2942  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468   × cxp 5141  dom cdm 5143  cres 5145  cfv 5926  (class class class)co 6690  t crest 16128  ∞Metcxmt 19779  Metcme 19780  fBascfbas 19782  filGencfg 19783  MetOpencmopn 19784  Topctop 20746  TopOnctopon 20763  Clsdccld 20868  clsccl 20870  Hauscha 21160  Filcfil 21696   fLim cflim 21785  CauFilccfil 23096  CMetcms 23098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ico 12219  df-icc 12220  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-haus 21167  df-fil 21697  df-flim 21790  df-cfil 23099  df-cmet 23101
This theorem is referenced by:  recmet  23166  cmsss  23193  bnsscmcl  27852  rrnheibor  33766
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