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Theorem bcth3 23174
 Description: Baire's Category Theorem, version 3: The intersection of countably many dense open sets is dense. (Contributed by Mario Carneiro, 10-Jan-2014.)
Hypothesis
Ref Expression
bcth.2 𝐽 = (MetOpen‘𝐷)
Assertion
Ref Expression
bcth3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋) → ((cls‘𝐽)‘ ran 𝑀) = 𝑋)
Distinct variable groups:   𝐷,𝑘   𝑘,𝐽   𝑘,𝑀   𝑘,𝑋

Proof of Theorem bcth3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cmetmet 23130 . . . . 5 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2 metxmet 22186 . . . . 5 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
31, 2syl 17 . . . 4 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
4 bcth.2 . . . . . . . . . 10 𝐽 = (MetOpen‘𝐷)
54mopntop 22292 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
65ad2antrr 762 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ Top)
7 ffvelrn 6397 . . . . . . . . . 10 ((𝑀:ℕ⟶𝐽𝑘 ∈ ℕ) → (𝑀𝑘) ∈ 𝐽)
8 elssuni 4499 . . . . . . . . . 10 ((𝑀𝑘) ∈ 𝐽 → (𝑀𝑘) ⊆ 𝐽)
97, 8syl 17 . . . . . . . . 9 ((𝑀:ℕ⟶𝐽𝑘 ∈ ℕ) → (𝑀𝑘) ⊆ 𝐽)
109adantll 750 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ⊆ 𝐽)
11 eqid 2651 . . . . . . . . 9 𝐽 = 𝐽
1211clsval2 20902 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑀𝑘) ⊆ 𝐽) → ((cls‘𝐽)‘(𝑀𝑘)) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))))
136, 10, 12syl2anc 694 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((cls‘𝐽)‘(𝑀𝑘)) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))))
144mopnuni 22293 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
1514ad2antrr 762 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝑋 = 𝐽)
1613, 15eqeq12d 2666 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 ↔ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽))
17 difeq2 3755 . . . . . . . 8 (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ( 𝐽 𝐽))
18 difid 3981 . . . . . . . 8 ( 𝐽 𝐽) = ∅
1917, 18syl6eq 2701 . . . . . . 7 (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ∅)
20 difss 3770 . . . . . . . . . . . 12 ( 𝐽 ∖ (𝑀𝑘)) ⊆ 𝐽
2111ntropn 20901 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ( 𝐽 ∖ (𝑀𝑘)) ⊆ 𝐽) → ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ∈ 𝐽)
226, 20, 21sylancl 695 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ∈ 𝐽)
23 elssuni 4499 . . . . . . . . . . 11 (((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ∈ 𝐽 → ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ⊆ 𝐽)
2422, 23syl 17 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ⊆ 𝐽)
25 dfss4 3891 . . . . . . . . . 10 (((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ⊆ 𝐽 ↔ ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))
2624, 25sylib 208 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))
27 id 22 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
28 elfvdm 6258 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
29 difexg 4841 . . . . . . . . . . . . . 14 (𝑋 ∈ dom ∞Met → (𝑋 ∖ (𝑀𝑘)) ∈ V)
3028, 29syl 17 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑘)) ∈ V)
3130adantr 480 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ (𝑀𝑘)) ∈ V)
32 fveq2 6229 . . . . . . . . . . . . . 14 (𝑥 = 𝑘 → (𝑀𝑥) = (𝑀𝑘))
3332difeq2d 3761 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (𝑋 ∖ (𝑀𝑥)) = (𝑋 ∖ (𝑀𝑘)))
34 eqid 2651 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))
3533, 34fvmptg 6319 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ (𝑋 ∖ (𝑀𝑘)) ∈ V) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = (𝑋 ∖ (𝑀𝑘)))
3627, 31, 35syl2anr 494 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = (𝑋 ∖ (𝑀𝑘)))
3715difeq1d 3760 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑋 ∖ (𝑀𝑘)) = ( 𝐽 ∖ (𝑀𝑘)))
3836, 37eqtrd 2685 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = ( 𝐽 ∖ (𝑀𝑘)))
3938fveq2d 6233 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))
4026, 39eqtr4d 2688 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)))
4140eqeq1d 2653 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4219, 41syl5ib 234 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4316, 42sylbid 230 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4443ralimdva 2991 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
453, 44sylan 487 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
46 ffvelrn 6397 . . . . . . . . 9 ((𝑀:ℕ⟶𝐽𝑥 ∈ ℕ) → (𝑀𝑥) ∈ 𝐽)
4714difeq1d 3760 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑥)) = ( 𝐽 ∖ (𝑀𝑥)))
4847adantr 480 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀𝑥)) = ( 𝐽 ∖ (𝑀𝑥)))
4911opncld 20885 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑀𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
505, 49sylan 487 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5148, 50eqeltrd 2730 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5246, 51sylan2 490 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀:ℕ⟶𝐽𝑥 ∈ ℕ)) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5352anassrs 681 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5453ralrimiva 2995 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
553, 54sylan 487 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5634fmpt 6421 . . . . 5 (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽) ↔ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽))
5755, 56sylib 208 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽))
58 nne 2827 . . . . . . 7 (¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅)
5958ralbii 3009 . . . . . 6 (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅)
60 ralnex 3021 . . . . . 6 (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅)
6159, 60bitr3i 266 . . . . 5 (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅)
624bcth 23172 . . . . . . 7 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) ≠ ∅) → ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅)
63623expia 1286 . . . . . 6 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽)) → (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) ≠ ∅ → ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅))
6463necon1bd 2841 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽)) → (¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ → ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅))
6561, 64syl5bi 232 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽)) → (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅ → ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅))
6657, 65syldan 486 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅ → ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅))
67 difeq2 3755 . . . . 5 (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ∅))
68 difexg 4841 . . . . . . . . . . . . . . . 16 (𝑋 ∈ dom ∞Met → (𝑋 ∖ (𝑀𝑥)) ∈ V)
6928, 68syl 17 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑥)) ∈ V)
7069ad2antrr 762 . . . . . . . . . . . . . 14 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀𝑥)) ∈ V)
7170ralrimiva 2995 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ V)
7234fnmpt 6058 . . . . . . . . . . . . 13 (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ V → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) Fn ℕ)
73 fniunfv 6545 . . . . . . . . . . . . 13 ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) Fn ℕ → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))
7471, 72, 733syl 18 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))
7536iuneq2dv 4574 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = 𝑘 ∈ ℕ (𝑋 ∖ (𝑀𝑘)))
7633cbviunv 4591 . . . . . . . . . . . . 13 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) = 𝑘 ∈ ℕ (𝑋 ∖ (𝑀𝑘))
7775, 76syl6eqr 2703 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)))
7874, 77eqtr3d 2687 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)))
79 iundif2 4619 . . . . . . . . . . 11 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) = (𝑋 𝑥 ∈ ℕ (𝑀𝑥))
8078, 79syl6eq 2701 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = (𝑋 𝑥 ∈ ℕ (𝑀𝑥)))
81 ffn 6083 . . . . . . . . . . . . 13 (𝑀:ℕ⟶𝐽𝑀 Fn ℕ)
8281adantl 481 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑀 Fn ℕ)
83 fniinfv 6296 . . . . . . . . . . . 12 (𝑀 Fn ℕ → 𝑥 ∈ ℕ (𝑀𝑥) = ran 𝑀)
8482, 83syl 17 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑥 ∈ ℕ (𝑀𝑥) = ran 𝑀)
8584difeq2d 3761 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 𝑥 ∈ ℕ (𝑀𝑥)) = (𝑋 ran 𝑀))
8614adantr 480 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑋 = 𝐽)
8786difeq1d 3760 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ran 𝑀) = ( 𝐽 ran 𝑀))
8880, 85, 873eqtrd 2689 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = ( 𝐽 ran 𝑀))
8988fveq2d 6233 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ((int‘𝐽)‘( 𝐽 ran 𝑀)))
9089difeq2d 3761 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
915adantr 480 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝐽 ∈ Top)
92 1nn 11069 . . . . . . . . 9 1 ∈ ℕ
93 biidd 252 . . . . . . . . . 10 (𝑘 = 1 → (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽) ↔ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽)))
94 fnfvelrn 6396 . . . . . . . . . . . . . 14 ((𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ ran 𝑀)
9582, 94sylan 487 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ ran 𝑀)
96 intss1 4524 . . . . . . . . . . . . 13 ((𝑀𝑘) ∈ ran 𝑀 ran 𝑀 ⊆ (𝑀𝑘))
9795, 96syl 17 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ran 𝑀 ⊆ (𝑀𝑘))
9897, 10sstrd 3646 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ran 𝑀 𝐽)
9998expcom 450 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽))
10093, 99vtoclga 3303 . . . . . . . . 9 (1 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽))
10192, 100ax-mp 5 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽)
10211clsval2 20902 . . . . . . . 8 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((cls‘𝐽)‘ ran 𝑀) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
10391, 101, 102syl2anc 694 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((cls‘𝐽)‘ ran 𝑀) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
10490, 103eqtr4d 2688 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ((cls‘𝐽)‘ ran 𝑀))
105 dif0 3983 . . . . . . 7 ( 𝐽 ∖ ∅) = 𝐽
10686, 105syl6reqr 2704 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ∅) = 𝑋)
107104, 106eqeq12d 2666 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ∅) ↔ ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
10867, 107syl5ib 234 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
1093, 108sylan 487 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
11045, 66, 1093syld 60 . 2 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
1111103impia 1280 1 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋) → ((cls‘𝐽)‘ ran 𝑀) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  ∅c0 3948  ∪ cuni 4468  ∩ cint 4507  ∪ ciun 4552  ∩ ciin 4553   ↦ cmpt 4762  dom cdm 5143  ran crn 5144   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  1c1 9975  ℕcn 11058  ∞Metcxmt 19779  Metcme 19780  MetOpencmopn 19784  Topctop 20746  Clsdccld 20868  intcnt 20869  clsccl 20870  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-inf2 8576  ax-dc 9306  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-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  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-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-lm 21081  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-cfil 23099  df-cau 23100  df-cmet 23101 This theorem is referenced by: (None)
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