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Theorem bcth3 23849
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 23804 . . . . 5 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2 metxmet 22859 . . . . 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 22965 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
65ad2antrr 722 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ Top)
7 ffvelrn 6845 . . . . . . . . . 10 ((𝑀:ℕ⟶𝐽𝑘 ∈ ℕ) → (𝑀𝑘) ∈ 𝐽)
8 elssuni 4866 . . . . . . . . . 10 ((𝑀𝑘) ∈ 𝐽 → (𝑀𝑘) ⊆ 𝐽)
97, 8syl 17 . . . . . . . . 9 ((𝑀:ℕ⟶𝐽𝑘 ∈ ℕ) → (𝑀𝑘) ⊆ 𝐽)
109adantll 710 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ⊆ 𝐽)
11 eqid 2826 . . . . . . . . 9 𝐽 = 𝐽
1211clsval2 21574 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑀𝑘) ⊆ 𝐽) → ((cls‘𝐽)‘(𝑀𝑘)) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))))
136, 10, 12syl2anc 584 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((cls‘𝐽)‘(𝑀𝑘)) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))))
144mopnuni 22966 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
1514ad2antrr 722 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝑋 = 𝐽)
1613, 15eqeq12d 2842 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 ↔ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽))
17 difeq2 4097 . . . . . . . 8 (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ( 𝐽 𝐽))
18 difid 4334 . . . . . . . 8 ( 𝐽 𝐽) = ∅
1917, 18syl6eq 2877 . . . . . . 7 (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ∅)
20 difss 4112 . . . . . . . . . . . 12 ( 𝐽 ∖ (𝑀𝑘)) ⊆ 𝐽
2111ntropn 21573 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ( 𝐽 ∖ (𝑀𝑘)) ⊆ 𝐽) → ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ∈ 𝐽)
226, 20, 21sylancl 586 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ∈ 𝐽)
23 elssuni 4866 . . . . . . . . . . 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 4239 . . . . . . . . . 10 (((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ⊆ 𝐽 ↔ ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))
2624, 25sylib 219 . . . . . . . . 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 6699 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
29 difexg 5228 . . . . . . . . . . . . . 14 (𝑋 ∈ dom ∞Met → (𝑋 ∖ (𝑀𝑘)) ∈ V)
3028, 29syl 17 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑘)) ∈ V)
3130adantr 481 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ (𝑀𝑘)) ∈ V)
32 fveq2 6667 . . . . . . . . . . . . . 14 (𝑥 = 𝑘 → (𝑀𝑥) = (𝑀𝑘))
3332difeq2d 4103 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (𝑋 ∖ (𝑀𝑥)) = (𝑋 ∖ (𝑀𝑘)))
34 eqid 2826 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))
3533, 34fvmptg 6763 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ (𝑋 ∖ (𝑀𝑘)) ∈ V) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = (𝑋 ∖ (𝑀𝑘)))
3627, 31, 35syl2anr 596 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = (𝑋 ∖ (𝑀𝑘)))
3715difeq1d 4102 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑋 ∖ (𝑀𝑘)) = ( 𝐽 ∖ (𝑀𝑘)))
3836, 37eqtrd 2861 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = ( 𝐽 ∖ (𝑀𝑘)))
3938fveq2d 6671 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))
4026, 39eqtr4d 2864 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)))
4140eqeq1d 2828 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4219, 41syl5ib 245 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4316, 42sylbid 241 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4443ralimdva 3182 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
453, 44sylan 580 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
46 ffvelrn 6845 . . . . . . . . 9 ((𝑀:ℕ⟶𝐽𝑥 ∈ ℕ) → (𝑀𝑥) ∈ 𝐽)
4714difeq1d 4102 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑥)) = ( 𝐽 ∖ (𝑀𝑥)))
4847adantr 481 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀𝑥)) = ( 𝐽 ∖ (𝑀𝑥)))
4911opncld 21557 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑀𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
505, 49sylan 580 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5148, 50eqeltrd 2918 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5246, 51sylan2 592 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀:ℕ⟶𝐽𝑥 ∈ ℕ)) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5352anassrs 468 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5453ralrimiva 3187 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
553, 54sylan 580 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5634fmpt 6870 . . . . 5 (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽) ↔ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽))
5755, 56sylib 219 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽))
58 nne 3025 . . . . . . 7 (¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅)
5958ralbii 3170 . . . . . 6 (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅)
60 ralnex 3241 . . . . . 6 (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅)
6159, 60bitr3i 278 . . . . 5 (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅)
624bcth 23847 . . . . . . 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 1115 . . . . . 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 3039 . . . . 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 243 . . . 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 591 . . 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 4097 . . . . 5 (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ∅))
68 difexg 5228 . . . . . . . . . . . . . . . 16 (𝑋 ∈ dom ∞Met → (𝑋 ∖ (𝑀𝑥)) ∈ V)
6928, 68syl 17 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑥)) ∈ V)
7069ad2antrr 722 . . . . . . . . . . . . . 14 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀𝑥)) ∈ V)
7170ralrimiva 3187 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ V)
7234fnmpt 6485 . . . . . . . . . . . . 13 (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ V → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) Fn ℕ)
73 fniunfv 7000 . . . . . . . . . . . . 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 4940 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = 𝑘 ∈ ℕ (𝑋 ∖ (𝑀𝑘)))
7633cbviunv 4962 . . . . . . . . . . . . 13 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) = 𝑘 ∈ ℕ (𝑋 ∖ (𝑀𝑘))
7775, 76syl6eqr 2879 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)))
7874, 77eqtr3d 2863 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)))
79 iundif2 4993 . . . . . . . . . . 11 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) = (𝑋 𝑥 ∈ ℕ (𝑀𝑥))
8078, 79syl6eq 2877 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = (𝑋 𝑥 ∈ ℕ (𝑀𝑥)))
81 ffn 6511 . . . . . . . . . . . . 13 (𝑀:ℕ⟶𝐽𝑀 Fn ℕ)
8281adantl 482 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑀 Fn ℕ)
83 fniinfv 6739 . . . . . . . . . . . 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 4103 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 𝑥 ∈ ℕ (𝑀𝑥)) = (𝑋 ran 𝑀))
8614adantr 481 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑋 = 𝐽)
8786difeq1d 4102 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ran 𝑀) = ( 𝐽 ran 𝑀))
8880, 85, 873eqtrd 2865 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = ( 𝐽 ran 𝑀))
8988fveq2d 6671 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ((int‘𝐽)‘( 𝐽 ran 𝑀)))
9089difeq2d 4103 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
915adantr 481 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝐽 ∈ Top)
92 1nn 11638 . . . . . . . . 9 1 ∈ ℕ
93 biidd 263 . . . . . . . . . 10 (𝑘 = 1 → (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽) ↔ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽)))
94 fnfvelrn 6844 . . . . . . . . . . . . . 14 ((𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ ran 𝑀)
9582, 94sylan 580 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ ran 𝑀)
96 intss1 4889 . . . . . . . . . . . . 13 ((𝑀𝑘) ∈ ran 𝑀 ran 𝑀 ⊆ (𝑀𝑘))
9795, 96syl 17 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ran 𝑀 ⊆ (𝑀𝑘))
9897, 10sstrd 3981 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ran 𝑀 𝐽)
9998expcom 414 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽))
10093, 99vtoclga 3579 . . . . . . . . 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 21574 . . . . . . . 8 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((cls‘𝐽)‘ ran 𝑀) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
10391, 101, 102syl2anc 584 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((cls‘𝐽)‘ ran 𝑀) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
10490, 103eqtr4d 2864 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ((cls‘𝐽)‘ ran 𝑀))
105 dif0 4336 . . . . . . 7 ( 𝐽 ∖ ∅) = 𝐽
10686, 105syl6reqr 2880 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ∅) = 𝑋)
107104, 106eqeq12d 2842 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ∅) ↔ ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
10867, 107syl5ib 245 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
1093, 108sylan 580 . . 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 1111 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 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  Vcvv 3500  cdif 3937  wss 3940  c0 4295   cuni 4837   cint 4874   ciun 4917   ciin 4918  cmpt 5143  dom cdm 5554  ran crn 5555   Fn wfn 6347  wf 6348  cfv 6352  1c1 10527  cn 11627  ∞Metcxmet 20446  Metcmet 20447  MetOpencmopn 20451  Topctop 21417  Clsdccld 21540  intcnt 21541  clsccl 21542  CMetccmet 23772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  ax-dc 9857  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-sup 8895  df-inf 8896  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-n0 11887  df-z 11971  df-uz 12233  df-q 12338  df-rp 12380  df-xneg 12497  df-xadd 12498  df-xmul 12499  df-ico 12734  df-rest 16686  df-topgen 16707  df-psmet 20453  df-xmet 20454  df-met 20455  df-bl 20456  df-mopn 20457  df-fbas 20458  df-fg 20459  df-top 21418  df-topon 21435  df-bases 21470  df-cld 21543  df-ntr 21544  df-cls 21545  df-nei 21622  df-lm 21753  df-fil 22370  df-fm 22462  df-flim 22463  df-flf 22464  df-cfil 23773  df-cau 23774  df-cmet 23775
This theorem is referenced by: (None)
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