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Theorem bcth3 23051
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 23007 . . . . 5 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2 metxmet 22062 . . . . 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 22168 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
65ad2antrr 761 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ Top)
7 ffvelrn 6318 . . . . . . . . . 10 ((𝑀:ℕ⟶𝐽𝑘 ∈ ℕ) → (𝑀𝑘) ∈ 𝐽)
8 elssuni 4438 . . . . . . . . . 10 ((𝑀𝑘) ∈ 𝐽 → (𝑀𝑘) ⊆ 𝐽)
97, 8syl 17 . . . . . . . . 9 ((𝑀:ℕ⟶𝐽𝑘 ∈ ℕ) → (𝑀𝑘) ⊆ 𝐽)
109adantll 749 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ⊆ 𝐽)
11 eqid 2621 . . . . . . . . 9 𝐽 = 𝐽
1211clsval2 20777 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑀𝑘) ⊆ 𝐽) → ((cls‘𝐽)‘(𝑀𝑘)) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))))
136, 10, 12syl2anc 692 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((cls‘𝐽)‘(𝑀𝑘)) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))))
144mopnuni 22169 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
1514ad2antrr 761 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝑋 = 𝐽)
1613, 15eqeq12d 2636 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 ↔ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽))
17 difeq2 3705 . . . . . . . 8 (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ( 𝐽 𝐽))
18 difid 3927 . . . . . . . 8 ( 𝐽 𝐽) = ∅
1917, 18syl6eq 2671 . . . . . . 7 (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ∅)
20 difss 3720 . . . . . . . . . . . 12 ( 𝐽 ∖ (𝑀𝑘)) ⊆ 𝐽
2111ntropn 20776 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ( 𝐽 ∖ (𝑀𝑘)) ⊆ 𝐽) → ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ∈ 𝐽)
226, 20, 21sylancl 693 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ∈ 𝐽)
23 elssuni 4438 . . . . . . . . . . 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 3841 . . . . . . . . . 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 6182 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
29 difexg 4773 . . . . . . . . . . . . . 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 6153 . . . . . . . . . . . . . 14 (𝑥 = 𝑘 → (𝑀𝑥) = (𝑀𝑘))
3332difeq2d 3711 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (𝑋 ∖ (𝑀𝑥)) = (𝑋 ∖ (𝑀𝑘)))
34 eqid 2621 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))
3533, 34fvmptg 6242 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ (𝑋 ∖ (𝑀𝑘)) ∈ V) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = (𝑋 ∖ (𝑀𝑘)))
3627, 31, 35syl2anr 495 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = (𝑋 ∖ (𝑀𝑘)))
3715difeq1d 3710 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑋 ∖ (𝑀𝑘)) = ( 𝐽 ∖ (𝑀𝑘)))
3836, 37eqtrd 2655 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = ( 𝐽 ∖ (𝑀𝑘)))
3938fveq2d 6157 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))
4026, 39eqtr4d 2658 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)))
4140eqeq1d 2623 . . . . . . 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 2957 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
453, 44sylan 488 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
46 ffvelrn 6318 . . . . . . . . 9 ((𝑀:ℕ⟶𝐽𝑥 ∈ ℕ) → (𝑀𝑥) ∈ 𝐽)
4714difeq1d 3710 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑥)) = ( 𝐽 ∖ (𝑀𝑥)))
4847adantr 481 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀𝑥)) = ( 𝐽 ∖ (𝑀𝑥)))
4911opncld 20760 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑀𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
505, 49sylan 488 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5148, 50eqeltrd 2698 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5246, 51sylan2 491 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀:ℕ⟶𝐽𝑥 ∈ ℕ)) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5352anassrs 679 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5453ralrimiva 2961 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
553, 54sylan 488 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5634fmpt 6342 . . . . 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 2794 . . . . . . 7 (¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅)
5958ralbii 2975 . . . . . 6 (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅)
60 ralnex 2987 . . . . . 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 23049 . . . . . . 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 1264 . . . . . 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 2808 . . . . 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 487 . . 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 3705 . . . . 5 (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ∅))
68 difexg 4773 . . . . . . . . . . . . . . . 16 (𝑋 ∈ dom ∞Met → (𝑋 ∖ (𝑀𝑥)) ∈ V)
6928, 68syl 17 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑥)) ∈ V)
7069ad2antrr 761 . . . . . . . . . . . . . 14 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀𝑥)) ∈ V)
7170ralrimiva 2961 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ V)
7234fnmpt 5982 . . . . . . . . . . . . 13 (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ V → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) Fn ℕ)
73 fniunfv 6465 . . . . . . . . . . . . 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 4513 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = 𝑘 ∈ ℕ (𝑋 ∖ (𝑀𝑘)))
7633cbviunv 4530 . . . . . . . . . . . . 13 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) = 𝑘 ∈ ℕ (𝑋 ∖ (𝑀𝑘))
7775, 76syl6eqr 2673 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)))
7874, 77eqtr3d 2657 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)))
79 iundif2 4558 . . . . . . . . . . 11 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) = (𝑋 𝑥 ∈ ℕ (𝑀𝑥))
8078, 79syl6eq 2671 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = (𝑋 𝑥 ∈ ℕ (𝑀𝑥)))
81 ffn 6007 . . . . . . . . . . . . 13 (𝑀:ℕ⟶𝐽𝑀 Fn ℕ)
8281adantl 482 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑀 Fn ℕ)
83 fniinfv 6219 . . . . . . . . . . . 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 3711 . . . . . . . . . 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 3710 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ran 𝑀) = ( 𝐽 ran 𝑀))
8880, 85, 873eqtrd 2659 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = ( 𝐽 ran 𝑀))
8988fveq2d 6157 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ((int‘𝐽)‘( 𝐽 ran 𝑀)))
9089difeq2d 3711 . . . . . . 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 10983 . . . . . . . . 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 6317 . . . . . . . . . . . . . 14 ((𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ ran 𝑀)
9582, 94sylan 488 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ ran 𝑀)
96 intss1 4462 . . . . . . . . . . . . 13 ((𝑀𝑘) ∈ ran 𝑀 ran 𝑀 ⊆ (𝑀𝑘))
9795, 96syl 17 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ran 𝑀 ⊆ (𝑀𝑘))
9897, 10sstrd 3597 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ran 𝑀 𝐽)
9998expcom 451 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽))
10093, 99vtoclga 3261 . . . . . . . . 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 20777 . . . . . . . 8 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((cls‘𝐽)‘ ran 𝑀) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
10391, 101, 102syl2anc 692 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((cls‘𝐽)‘ ran 𝑀) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
10490, 103eqtr4d 2658 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ((cls‘𝐽)‘ ran 𝑀))
105 dif0 3929 . . . . . . 7 ( 𝐽 ∖ ∅) = 𝐽
10686, 105syl6reqr 2674 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ∅) = 𝑋)
107104, 106eqeq12d 2636 . . . . 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 488 . . 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 1258 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3189  cdif 3556  wss 3559  c0 3896   cuni 4407   cint 4445   ciun 4490   ciin 4491  cmpt 4678  dom cdm 5079  ran crn 5080   Fn wfn 5847  wf 5848  cfv 5852  1c1 9889  cn 10972  ∞Metcxmt 19663  Metcme 19664  MetOpencmopn 19668  Topctop 20630  Clsdccld 20743  intcnt 20744  clsccl 20745  CMetcms 22975
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-dc 9220  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-er 7694  df-map 7811  df-pm 7812  df-en 7908  df-dom 7909  df-sdom 7910  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-q 11741  df-rp 11785  df-xneg 11898  df-xadd 11899  df-xmul 11900  df-ico 12131  df-rest 16015  df-topgen 16036  df-psmet 19670  df-xmet 19671  df-met 19672  df-bl 19673  df-mopn 19674  df-fbas 19675  df-fg 19676  df-top 20631  df-topon 20648  df-bases 20674  df-cld 20746  df-ntr 20747  df-cls 20748  df-nei 20825  df-lm 20956  df-fil 21573  df-fm 21665  df-flim 21666  df-flf 21667  df-cfil 22976  df-cau 22977  df-cmet 22978
This theorem is referenced by: (None)
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