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Theorem bcth3 23411
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 23366 . . . . 5 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2 metxmet 22421 . . . . 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 22527 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
65ad2antrr 717 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ Top)
7 ffvelrn 6549 . . . . . . . . . 10 ((𝑀:ℕ⟶𝐽𝑘 ∈ ℕ) → (𝑀𝑘) ∈ 𝐽)
8 elssuni 4627 . . . . . . . . . 10 ((𝑀𝑘) ∈ 𝐽 → (𝑀𝑘) ⊆ 𝐽)
97, 8syl 17 . . . . . . . . 9 ((𝑀:ℕ⟶𝐽𝑘 ∈ ℕ) → (𝑀𝑘) ⊆ 𝐽)
109adantll 705 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ⊆ 𝐽)
11 eqid 2765 . . . . . . . . 9 𝐽 = 𝐽
1211clsval2 21137 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑀𝑘) ⊆ 𝐽) → ((cls‘𝐽)‘(𝑀𝑘)) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))))
136, 10, 12syl2anc 579 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((cls‘𝐽)‘(𝑀𝑘)) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))))
144mopnuni 22528 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
1514ad2antrr 717 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝑋 = 𝐽)
1613, 15eqeq12d 2780 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 ↔ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽))
17 difeq2 3886 . . . . . . . 8 (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ( 𝐽 𝐽))
18 difid 4115 . . . . . . . 8 ( 𝐽 𝐽) = ∅
1917, 18syl6eq 2815 . . . . . . 7 (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ∅)
20 difss 3901 . . . . . . . . . . . 12 ( 𝐽 ∖ (𝑀𝑘)) ⊆ 𝐽
2111ntropn 21136 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ( 𝐽 ∖ (𝑀𝑘)) ⊆ 𝐽) → ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ∈ 𝐽)
226, 20, 21sylancl 580 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ∈ 𝐽)
23 elssuni 4627 . . . . . . . . . . 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 4025 . . . . . . . . . 10 (((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))) ⊆ 𝐽 ↔ ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))
2624, 25sylib 209 . . . . . . . . 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 6409 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
29 difexg 4971 . . . . . . . . . . . . . 14 (𝑋 ∈ dom ∞Met → (𝑋 ∖ (𝑀𝑘)) ∈ V)
3028, 29syl 17 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑘)) ∈ V)
3130adantr 472 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ (𝑀𝑘)) ∈ V)
32 fveq2 6377 . . . . . . . . . . . . . 14 (𝑥 = 𝑘 → (𝑀𝑥) = (𝑀𝑘))
3332difeq2d 3892 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (𝑋 ∖ (𝑀𝑥)) = (𝑋 ∖ (𝑀𝑘)))
34 eqid 2765 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))
3533, 34fvmptg 6471 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ (𝑋 ∖ (𝑀𝑘)) ∈ V) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = (𝑋 ∖ (𝑀𝑘)))
3627, 31, 35syl2anr 590 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = (𝑋 ∖ (𝑀𝑘)))
3715difeq1d 3891 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑋 ∖ (𝑀𝑘)) = ( 𝐽 ∖ (𝑀𝑘)))
3836, 37eqtrd 2799 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = ( 𝐽 ∖ (𝑀𝑘)))
3938fveq2d 6381 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))
4026, 39eqtr4d 2802 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)))
4140eqeq1d 2767 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (( 𝐽 ∖ ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘))))) = ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4219, 41syl5ib 235 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ∖ (𝑀𝑘)))) = 𝐽 → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4316, 42sylbid 231 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
4443ralimdva 3109 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
453, 44sylan 575 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅))
46 ffvelrn 6549 . . . . . . . . 9 ((𝑀:ℕ⟶𝐽𝑥 ∈ ℕ) → (𝑀𝑥) ∈ 𝐽)
4714difeq1d 3891 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑥)) = ( 𝐽 ∖ (𝑀𝑥)))
4847adantr 472 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀𝑥)) = ( 𝐽 ∖ (𝑀𝑥)))
4911opncld 21120 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑀𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
505, 49sylan 575 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5148, 50eqeltrd 2844 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5246, 51sylan2 586 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀:ℕ⟶𝐽𝑥 ∈ ℕ)) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5352anassrs 459 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5453ralrimiva 3113 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
553, 54sylan 575 . . . . 5 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽))
5634fmpt 6572 . . . . 5 (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ (Clsd‘𝐽) ↔ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽))
5755, 56sylib 209 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))):ℕ⟶(Clsd‘𝐽))
58 nne 2941 . . . . . . 7 (¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅)
5958ralbii 3127 . . . . . 6 (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅)
60 ralnex 3139 . . . . . 6 (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅)
6159, 60bitr3i 268 . . . . 5 (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) = ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘)) ≠ ∅)
624bcth 23409 . . . . . . 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 1150 . . . . . 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 2955 . . . . 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 233 . . . 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 585 . . 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 3886 . . . . 5 (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ∅))
68 difexg 4971 . . . . . . . . . . . . . . . 16 (𝑋 ∈ dom ∞Met → (𝑋 ∖ (𝑀𝑥)) ∈ V)
6928, 68syl 17 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀𝑥)) ∈ V)
7069ad2antrr 717 . . . . . . . . . . . . . 14 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀𝑥)) ∈ V)
7170ralrimiva 3113 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ V)
7234fnmpt 6200 . . . . . . . . . . . . 13 (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) ∈ V → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) Fn ℕ)
73 fniunfv 6699 . . . . . . . . . . . . 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 4700 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = 𝑘 ∈ ℕ (𝑋 ∖ (𝑀𝑘)))
7633cbviunv 4717 . . . . . . . . . . . . 13 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) = 𝑘 ∈ ℕ (𝑋 ∖ (𝑀𝑘))
7775, 76syl6eqr 2817 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))‘𝑘) = 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)))
7874, 77eqtr3d 2801 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)))
79 iundif2 4745 . . . . . . . . . . 11 𝑥 ∈ ℕ (𝑋 ∖ (𝑀𝑥)) = (𝑋 𝑥 ∈ ℕ (𝑀𝑥))
8078, 79syl6eq 2815 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = (𝑋 𝑥 ∈ ℕ (𝑀𝑥)))
81 ffn 6225 . . . . . . . . . . . . 13 (𝑀:ℕ⟶𝐽𝑀 Fn ℕ)
8281adantl 473 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑀 Fn ℕ)
83 fniinfv 6448 . . . . . . . . . . . 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 3892 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 𝑥 ∈ ℕ (𝑀𝑥)) = (𝑋 ran 𝑀))
8614adantr 472 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑋 = 𝐽)
8786difeq1d 3891 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ran 𝑀) = ( 𝐽 ran 𝑀))
8880, 85, 873eqtrd 2803 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))) = ( 𝐽 ran 𝑀))
8988fveq2d 6381 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ((int‘𝐽)‘( 𝐽 ran 𝑀)))
9089difeq2d 3892 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
915adantr 472 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝐽 ∈ Top)
92 1nn 11289 . . . . . . . . 9 1 ∈ ℕ
93 biidd 253 . . . . . . . . . 10 (𝑘 = 1 → (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽) ↔ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽)))
94 fnfvelrn 6548 . . . . . . . . . . . . . 14 ((𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ ran 𝑀)
9582, 94sylan 575 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ ran 𝑀)
96 intss1 4650 . . . . . . . . . . . . 13 ((𝑀𝑘) ∈ ran 𝑀 ran 𝑀 ⊆ (𝑀𝑘))
9795, 96syl 17 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ran 𝑀 ⊆ (𝑀𝑘))
9897, 10sstrd 3773 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ran 𝑀 𝐽)
9998expcom 402 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ran 𝑀 𝐽))
10093, 99vtoclga 3425 . . . . . . . . 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 21137 . . . . . . . 8 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((cls‘𝐽)‘ ran 𝑀) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
10391, 101, 102syl2anc 579 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((cls‘𝐽)‘ ran 𝑀) = ( 𝐽 ∖ ((int‘𝐽)‘( 𝐽 ran 𝑀))))
10490, 103eqtr4d 2802 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ((cls‘𝐽)‘ ran 𝑀))
105 dif0 4117 . . . . . . 7 ( 𝐽 ∖ ∅) = 𝐽
10686, 105syl6reqr 2818 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ( 𝐽 ∖ ∅) = 𝑋)
107104, 106eqeq12d 2780 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (( 𝐽 ∖ ((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥))))) = ( 𝐽 ∖ ∅) ↔ ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
10867, 107syl5ib 235 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀𝑥)))) = ∅ → ((cls‘𝐽)‘ ran 𝑀) = 𝑋))
1093, 108sylan 575 . . 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 1145 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 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  cdif 3731  wss 3734  c0 4081   cuni 4596   cint 4635   ciun 4678   ciin 4679  cmpt 4890  dom cdm 5279  ran crn 5280   Fn wfn 6065  wf 6066  cfv 6070  1c1 10192  cn 11276  ∞Metcxmet 20007  Metcmet 20008  MetOpencmopn 20012  Topctop 20980  Clsdccld 21103  intcnt 21104  clsccl 21105  CMetccmet 23334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-inf2 8755  ax-dc 9523  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-er 7949  df-map 8064  df-pm 8065  df-en 8163  df-dom 8164  df-sdom 8165  df-sup 8557  df-inf 8558  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-n0 11541  df-z 11627  df-uz 11890  df-q 11993  df-rp 12032  df-xneg 12149  df-xadd 12150  df-xmul 12151  df-ico 12386  df-rest 16352  df-topgen 16373  df-psmet 20014  df-xmet 20015  df-met 20016  df-bl 20017  df-mopn 20018  df-fbas 20019  df-fg 20020  df-top 20981  df-topon 20998  df-bases 21033  df-cld 21106  df-ntr 21107  df-cls 21108  df-nei 21185  df-lm 21316  df-fil 21932  df-fm 22024  df-flim 22025  df-flf 22026  df-cfil 23335  df-cau 23336  df-cmet 23337
This theorem is referenced by: (None)
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