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Theorem bcth3 24695
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 24650 . . . . 5 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2 metxmet 23687 . . . . 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 23793 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
65ad2antrr 724 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ Top)
7 ffvelcdm 7032 . . . . . . . . . 10 ((𝑀:ℕ⟶𝐽𝑘 ∈ ℕ) → (𝑀𝑘) ∈ 𝐽)
8 elssuni 4898 . . . . . . . . . 10 ((𝑀𝑘) ∈ 𝐽 → (𝑀𝑘) ⊆ 𝐽)
97, 8syl 17 . . . . . . . . 9 ((𝑀:ℕ⟶𝐽𝑘 ∈ ℕ) → (𝑀𝑘) ⊆ 𝐽)
109adantll 712 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ⊆ 𝐽)
11 eqid 2736 . . . . . . . . 9 𝐽 = 𝐽
1211clsval2 22401 . . . . . . . 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 23794 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
1514ad2antrr 724 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝑋 = 𝐽)
1613, 15eqeq12d 2752 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 ↔ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽))
17 difeq2 4076 . . . . . . . 8 (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ( 𝐽 𝐽))
18 difid 4330 . . . . . . . 8 ( 𝐽 𝐽) = ∅
1917, 18eqtrdi 2792 . . . . . . 7 (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ∅)
20 difss 4091 . . . . . . . . . . . 12 ( 𝐽 ∖ (𝑀𝑘)) ⊆ 𝐽
2111ntropn 22400 . . . . . . . . . . . 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 4898 . . . . . . . . . . 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 4218 . . . . . . . . . 10 (((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ⊆ 𝐽 ↔ ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))
2624, 25sylib 217 . . . . . . . . 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 6879 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2928difexd 5286 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑘)) ∈ V)
3029adantr 481 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ (𝑀𝑘)) ∈ V)
31 fveq2 6842 . . . . . . . . . . . . . 14 (𝑥 = 𝑘 → (𝑀𝑥) = (𝑀𝑘))
3231difeq2d 4082 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (𝑋 ∖ (𝑀𝑥)) = (𝑋 ∖ (𝑀𝑘)))
33 eqid 2736 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))
3432, 33fvmptg 6946 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ (𝑋 ∖ (𝑀𝑘)) ∈ V) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = (𝑋 ∖ (𝑀𝑘)))
3527, 30, 34syl2anr 597 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = (𝑋 ∖ (𝑀𝑘)))
3615difeq1d 4081 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑋 ∖ (𝑀𝑘)) = ( 𝐽 ∖ (𝑀𝑘)))
3735, 36eqtrd 2776 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = ( 𝐽 ∖ (𝑀𝑘)))
3837fveq2d 6846 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))
3926, 38eqtr4d 2779 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)))
4039eqeq1d 2738 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4119, 40imbitrid 243 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4216, 41sylbid 239 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4342ralimdva 3164 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
443, 43sylan 580 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
45 ffvelcdm 7032 . . . . . . . . 9 ((𝑀:ℕ⟶𝐽𝑥 ∈ ℕ) → (𝑀𝑥) ∈ 𝐽)
4614difeq1d 4081 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑥)) = ( 𝐽 ∖ (𝑀𝑥)))
4746adantr 481 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀𝑥)) = ( 𝐽 ∖ (𝑀𝑥)))
4811opncld 22384 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑀𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
495, 48sylan 580 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5047, 49eqeltrd 2838 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5145, 50sylan2 593 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀:ℕ⟶𝐽𝑥 ∈ ℕ)) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5251anassrs 468 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5352ralrimiva 3143 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
543, 53sylan 580 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5533fmpt 7058 . . . . 5 (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽) ↔ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽))
5654, 55sylib 217 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽))
57 nne 2947 . . . . . . 7 (¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅)
5857ralbii 3096 . . . . . 6 (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅)
59 ralnex 3075 . . . . . 6 (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅)
6058, 59bitr3i 276 . . . . 5 (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅)
614bcth 24693 . . . . . . 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)) ≠ ∅)
62613expia 1121 . . . . . 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)) ≠ ∅))
6362necon1bd 2961 . . . . 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)))) = ∅))
6460, 63biimtrid 241 . . . 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)))) = ∅))
6556, 64syldan 591 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅ → ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅))
66 difeq2 4076 . . . . 5 (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ∅))
6728difexd 5286 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑥)) ∈ V)
6867ad2antrr 724 . . . . . . . . . . . . . 14 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀𝑥)) ∈ V)
6968ralrimiva 3143 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ V)
7033fnmpt 6641 . . . . . . . . . . . . 13 (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ V → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) Fn ℕ)
71 fniunfv 7194 . . . . . . . . . . . . 13 ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) Fn ℕ → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))
7269, 70, 713syl 18 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))
7335iuneq2dv 4978 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = 𝑘 ∈ ℕ (𝑋 ∖ (𝑀𝑘)))
7432cbviunv 5000 . . . . . . . . . . . . 13 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) = 𝑘 ∈ ℕ (𝑋 ∖ (𝑀𝑘))
7573, 74eqtr4di 2794 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)))
7672, 75eqtr3d 2778 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)))
77 iundif2 5034 . . . . . . . . . . 11 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) = (𝑋 𝑥 ∈ ℕ (𝑀𝑥))
7876, 77eqtrdi 2792 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = (𝑋 𝑥 ∈ ℕ (𝑀𝑥)))
79 ffn 6668 . . . . . . . . . . . . 13 (𝑀:ℕ⟶𝐽𝑀 Fn ℕ)
8079adantl 482 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑀 Fn ℕ)
81 fniinfv 6919 . . . . . . . . . . . 12 (𝑀 Fn ℕ → 𝑥 ∈ ℕ (𝑀𝑥) = ran 𝑀)
8280, 81syl 17 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑥 ∈ ℕ (𝑀𝑥) = ran 𝑀)
8382difeq2d 4082 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 𝑥 ∈ ℕ (𝑀𝑥)) = (𝑋 ran 𝑀))
8414adantr 481 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑋 = 𝐽)
8584difeq1d 4081 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ran 𝑀) = ( 𝐽 ran 𝑀))
8678, 83, 853eqtrd 2780 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = ( 𝐽 ran 𝑀))
8786fveq2d 6846 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ((int‘𝐽)‘( 𝐽 ran 𝑀)))
8887difeq2d 4082 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
895adantr 481 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝐽 ∈ Top)
90 1nn 12164 . . . . . . . . 9 1 ∈ ℕ
91 biidd 261 . . . . . . . . . 10 (𝑘 = 1 → (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽) ↔ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽)))
92 fnfvelrn 7031 . . . . . . . . . . . . . 14 ((𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ ran 𝑀)
9380, 92sylan 580 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ ran 𝑀)
94 intss1 4924 . . . . . . . . . . . . 13 ((𝑀𝑘) ∈ ran 𝑀 ran 𝑀 ⊆ (𝑀𝑘))
9593, 94syl 17 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ran 𝑀 ⊆ (𝑀𝑘))
9695, 10sstrd 3954 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ran 𝑀 𝐽)
9796expcom 414 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽))
9891, 97vtoclga 3534 . . . . . . . . 9 (1 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽))
9990, 98ax-mp 5 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽)
10011clsval2 22401 . . . . . . . 8 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((cls‘𝐽)‘ ran 𝑀) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
10189, 99, 100syl2anc 584 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((cls‘𝐽)‘ ran 𝑀) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
10288, 101eqtr4d 2779 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ((cls‘𝐽)‘ ran 𝑀))
103 dif0 4332 . . . . . . 7 ( 𝐽 ∖ ∅) = 𝐽
104103, 84eqtr4id 2795 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ∅) = 𝑋)
105102, 104eqeq12d 2752 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ∅) ↔ ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
10666, 105imbitrid 243 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
1073, 106sylan 580 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
10844, 65, 1073syld 60 . 2 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
1091083impia 1117 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 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  wss 3910  c0 4282   cuni 4865   cint 4907   ciun 4954   ciin 4955  cmpt 5188  dom cdm 5633  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  1c1 11052  cn 12153  ∞Metcxmet 20781  Metcmet 20782  MetOpencmopn 20786  Topctop 22242  Clsdccld 22367  intcnt 22368  clsccl 22369  CMetccmet 24618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-dc 10382  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ico 13270  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-top 22243  df-topon 22260  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lm 22580  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-cfil 24619  df-cau 24620  df-cmet 24621
This theorem is referenced by: (None)
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