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Theorem vitalilem3 23185
 Description: Lemma for vitali 23188. (Contributed by Mario Carneiro, 16-Jun-2014.)
Hypotheses
Ref Expression
vitali.1 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
vitali.2 𝑆 = ((0[,]1) / )
vitali.3 (𝜑𝐹 Fn 𝑆)
vitali.4 (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
vitali.5 (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
vitali.6 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})
vitali.7 (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
Assertion
Ref Expression
vitalilem3 (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))
Distinct variable groups:   𝑚,𝑛,𝑠,𝑥,𝑦,𝑧,𝐺   𝜑,𝑚,𝑛,𝑥,𝑧   𝑧,𝑆   𝑇,𝑚,𝑥   𝑚,𝐹,𝑛,𝑠,𝑥,𝑦,𝑧   ,𝑚,𝑛,𝑠,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑠)   𝑆(𝑥,𝑦,𝑚,𝑛,𝑠)   𝑇(𝑦,𝑧,𝑛,𝑠)

Proof of Theorem vitalilem3
Dummy variables 𝑘 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprlr 799 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ (𝑇𝑚))
2 simprll 798 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑚 ∈ ℕ)
3 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
43oveq2d 6565 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑚)))
54eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹))
65rabbidv 3164 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
7 vitali.6 . . . . . . . . . . . . . 14 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})
8 reex 9906 . . . . . . . . . . . . . . 15 ℝ ∈ V
98rabex 4740 . . . . . . . . . . . . . 14 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ∈ V
106, 7, 9fvmpt 6191 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
112, 10syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
121, 11eleqtrd 2690 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
13 oveq1 6556 . . . . . . . . . . . . 13 (𝑠 = 𝑤 → (𝑠 − (𝐺𝑚)) = (𝑤 − (𝐺𝑚)))
1413eleq1d 2672 . . . . . . . . . . . 12 (𝑠 = 𝑤 → ((𝑠 − (𝐺𝑚)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1514elrab 3331 . . . . . . . . . . 11 (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1612, 15sylib 207 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1716simpld 474 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ ℝ)
1817recnd 9947 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ ℂ)
19 vitali.5 . . . . . . . . . . . . 13 (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
20 f1of 6050 . . . . . . . . . . . . 13 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
2119, 20syl 17 . . . . . . . . . . . 12 (𝜑𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
22 inss1 3795 . . . . . . . . . . . 12 (ℚ ∩ (-1[,]1)) ⊆ ℚ
23 fss 5969 . . . . . . . . . . . 12 ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℚ) → 𝐺:ℕ⟶ℚ)
2421, 22, 23sylancl 693 . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶ℚ)
2524adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ⟶ℚ)
2625, 2ffvelrnd 6268 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) ∈ ℚ)
27 qcn 11678 . . . . . . . . 9 ((𝐺𝑚) ∈ ℚ → (𝐺𝑚) ∈ ℂ)
2826, 27syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) ∈ ℂ)
29 simprrl 800 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑘 ∈ ℕ)
3025, 29ffvelrnd 6268 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑘) ∈ ℚ)
31 qcn 11678 . . . . . . . . 9 ((𝐺𝑘) ∈ ℚ → (𝐺𝑘) ∈ ℂ)
3230, 31syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑘) ∈ ℂ)
33 vitali.1 . . . . . . . . . . . . 13 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
3433vitalilem1 23182 . . . . . . . . . . . 12 Er (0[,]1)
3534a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → Er (0[,]1))
36 vitali.2 . . . . . . . . . . . . . . . . 17 𝑆 = ((0[,]1) / )
37 vitali.3 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 Fn 𝑆)
38 vitali.4 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
39 vitali.7 . . . . . . . . . . . . . . . . 17 (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
4033, 36, 37, 38, 19, 7, 39vitalilem2 23184 . . . . . . . . . . . . . . . 16 (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))
4140simp1d 1066 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ⊆ (0[,]1))
4241adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ran 𝐹 ⊆ (0[,]1))
4316simprd 478 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) ∈ ran 𝐹)
4442, 43sseldd 3569 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) ∈ (0[,]1))
45 simprrr 801 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ (𝑇𝑘))
46 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
4746oveq2d 6565 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑘)))
4847eleq1d 2672 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑘 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹))
4948rabbidv 3164 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
508rabex 4740 . . . . . . . . . . . . . . . . . . 19 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹} ∈ V
5149, 7, 50fvmpt 6191 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → (𝑇𝑘) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
5229, 51syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑇𝑘) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
5345, 52eleqtrd 2690 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
54 oveq1 6556 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑤 → (𝑠 − (𝐺𝑘)) = (𝑤 − (𝐺𝑘)))
5554eleq1d 2672 . . . . . . . . . . . . . . . . 17 (𝑠 = 𝑤 → ((𝑠 − (𝐺𝑘)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5655elrab 3331 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5753, 56sylib 207 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5857simprd 478 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑘)) ∈ ran 𝐹)
5942, 58sseldd 3569 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑘)) ∈ (0[,]1))
6044, 59jca 553 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) ∈ (0[,]1) ∧ (𝑤 − (𝐺𝑘)) ∈ (0[,]1)))
6118, 28, 32nnncan1d 10305 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) = ((𝐺𝑘) − (𝐺𝑚)))
62 qsubcl 11683 . . . . . . . . . . . . . 14 (((𝐺𝑘) ∈ ℚ ∧ (𝐺𝑚) ∈ ℚ) → ((𝐺𝑘) − (𝐺𝑚)) ∈ ℚ)
6330, 26, 62syl2anc 691 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝐺𝑘) − (𝐺𝑚)) ∈ ℚ)
6461, 63eqeltrd 2688 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ)
65 oveq12 6558 . . . . . . . . . . . . . 14 ((𝑥 = (𝑤 − (𝐺𝑚)) ∧ 𝑦 = (𝑤 − (𝐺𝑘))) → (𝑥𝑦) = ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))))
6665eleq1d 2672 . . . . . . . . . . . . 13 ((𝑥 = (𝑤 − (𝐺𝑚)) ∧ 𝑦 = (𝑤 − (𝐺𝑘))) → ((𝑥𝑦) ∈ ℚ ↔ ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ))
6766, 33brab2ga 5117 . . . . . . . . . . . 12 ((𝑤 − (𝐺𝑚)) (𝑤 − (𝐺𝑘)) ↔ (((𝑤 − (𝐺𝑚)) ∈ (0[,]1) ∧ (𝑤 − (𝐺𝑘)) ∈ (0[,]1)) ∧ ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ))
6860, 64, 67sylanbrc 695 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) (𝑤 − (𝐺𝑘)))
6935, 68erthi 7680 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → [(𝑤 − (𝐺𝑚))] = [(𝑤 − (𝐺𝑘))] )
7069fveq2d 6107 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝐹‘[(𝑤 − (𝐺𝑘))] ))
71 fveq2 6103 . . . . . . . . . . . . . . . . 17 ([𝑣] = 𝑤 → (𝐹‘[𝑣] ) = (𝐹𝑤))
7271eceq1d 7670 . . . . . . . . . . . . . . . 16 ([𝑣] = 𝑤 → [(𝐹‘[𝑣] )] = [(𝐹𝑤)] )
7372fveq2d 6107 . . . . . . . . . . . . . . 15 ([𝑣] = 𝑤 → (𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[(𝐹𝑤)] ))
7473, 71eqeq12d 2625 . . . . . . . . . . . . . 14 ([𝑣] = 𝑤 → ((𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[𝑣] ) ↔ (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
7534a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 ∈ (0[,]1)) → Er (0[,]1))
76 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . 23 (0[,]1) ∈ V
77 erex 7653 . . . . . . . . . . . . . . . . . . . . . . 23 ( Er (0[,]1) → ((0[,]1) ∈ V → ∈ V))
7834, 76, 77mp2 9 . . . . . . . . . . . . . . . . . . . . . 22 ∈ V
7978ecelqsi 7690 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ (0[,]1) → [𝑣] ∈ ((0[,]1) / ))
8079, 36syl6eleqr 2699 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ (0[,]1) → [𝑣] 𝑆)
8180adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] 𝑆)
8238adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 ∈ (0[,]1)) → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
83 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 ∈ (0[,]1))
84 erdm 7639 . . . . . . . . . . . . . . . . . . . . . . 23 ( Er (0[,]1) → dom = (0[,]1))
8534, 84ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 dom = (0[,]1)
8685eleq2i 2680 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ dom 𝑣 ∈ (0[,]1))
87 ecdmn0 7676 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ dom ↔ [𝑣] ≠ ∅)
8886, 87bitr3i 265 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ (0[,]1) ↔ [𝑣] ≠ ∅)
8983, 88sylib 207 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] ≠ ∅)
90 neeq1 2844 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = [𝑣] → (𝑧 ≠ ∅ ↔ [𝑣] ≠ ∅))
91 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = [𝑣] → (𝐹𝑧) = (𝐹‘[𝑣] ))
92 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = [𝑣] 𝑧 = [𝑣] )
9391, 92eleq12d 2682 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = [𝑣] → ((𝐹𝑧) ∈ 𝑧 ↔ (𝐹‘[𝑣] ) ∈ [𝑣] ))
9490, 93imbi12d 333 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = [𝑣] → ((𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧) ↔ ([𝑣] ≠ ∅ → (𝐹‘[𝑣] ) ∈ [𝑣] )))
9594rspcv 3278 . . . . . . . . . . . . . . . . . . 19 ([𝑣] 𝑆 → (∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧) → ([𝑣] ≠ ∅ → (𝐹‘[𝑣] ) ∈ [𝑣] )))
9681, 82, 89, 95syl3c 64 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ) ∈ [𝑣] )
97 fvex 6113 . . . . . . . . . . . . . . . . . . 19 (𝐹‘[𝑣] ) ∈ V
98 vex 3176 . . . . . . . . . . . . . . . . . . 19 𝑣 ∈ V
9997, 98elec 7673 . . . . . . . . . . . . . . . . . 18 ((𝐹‘[𝑣] ) ∈ [𝑣] 𝑣 (𝐹‘[𝑣] ))
10096, 99sylib 207 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 (𝐹‘[𝑣] ))
10175, 100erthi 7680 . . . . . . . . . . . . . . . 16 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] = [(𝐹‘[𝑣] )] )
102101eqcomd 2616 . . . . . . . . . . . . . . 15 ((𝜑𝑣 ∈ (0[,]1)) → [(𝐹‘[𝑣] )] = [𝑣] )
103102fveq2d 6107 . . . . . . . . . . . . . 14 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[𝑣] ))
10436, 74, 103ectocld 7701 . . . . . . . . . . . . 13 ((𝜑𝑤𝑆) → (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤))
105104ralrimiva 2949 . . . . . . . . . . . 12 (𝜑 → ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤))
106 eceq1 7669 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑤) → [𝑧] = [(𝐹𝑤)] )
107106fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑤) → (𝐹‘[𝑧] ) = (𝐹‘[(𝐹𝑤)] ))
108 id 22 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑤) → 𝑧 = (𝐹𝑤))
109107, 108eqeq12d 2625 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑤) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
110109ralrn 6270 . . . . . . . . . . . . 13 (𝐹 Fn 𝑆 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 ↔ ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
11137, 110syl 17 . . . . . . . . . . . 12 (𝜑 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 ↔ ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
112105, 111mpbird 246 . . . . . . . . . . 11 (𝜑 → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧)
113112adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧)
114 eceq1 7669 . . . . . . . . . . . . 13 (𝑧 = (𝑤 − (𝐺𝑚)) → [𝑧] = [(𝑤 − (𝐺𝑚))] )
115114fveq2d 6107 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑚)) → (𝐹‘[𝑧] ) = (𝐹‘[(𝑤 − (𝐺𝑚))] ))
116 id 22 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑚)) → 𝑧 = (𝑤 − (𝐺𝑚)))
117115, 116eqeq12d 2625 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑚)) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝑤 − (𝐺𝑚))))
118117rspcv 3278 . . . . . . . . . 10 ((𝑤 − (𝐺𝑚)) ∈ ran 𝐹 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 → (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝑤 − (𝐺𝑚))))
11943, 113, 118sylc 63 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝑤 − (𝐺𝑚)))
120 eceq1 7669 . . . . . . . . . . . . 13 (𝑧 = (𝑤 − (𝐺𝑘)) → [𝑧] = [(𝑤 − (𝐺𝑘))] )
121120fveq2d 6107 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑘)) → (𝐹‘[𝑧] ) = (𝐹‘[(𝑤 − (𝐺𝑘))] ))
122 id 22 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑘)) → 𝑧 = (𝑤 − (𝐺𝑘)))
123121, 122eqeq12d 2625 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑘)) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺𝑘))] ) = (𝑤 − (𝐺𝑘))))
124123rspcv 3278 . . . . . . . . . 10 ((𝑤 − (𝐺𝑘)) ∈ ran 𝐹 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 → (𝐹‘[(𝑤 − (𝐺𝑘))] ) = (𝑤 − (𝐺𝑘))))
12558, 113, 124sylc 63 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑘))] ) = (𝑤 − (𝐺𝑘)))
12670, 119, 1253eqtr3d 2652 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) = (𝑤 − (𝐺𝑘)))
12718, 28, 32, 126subcand 10312 . . . . . . 7 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) = (𝐺𝑘))
12819adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
129 f1of1 6049 . . . . . . . . 9 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
130128, 129syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
131 f1fveq 6420 . . . . . . . 8 ((𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)) ∧ (𝑚 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → ((𝐺𝑚) = (𝐺𝑘) ↔ 𝑚 = 𝑘))
132130, 2, 29, 131syl12anc 1316 . . . . . . 7 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝐺𝑚) = (𝐺𝑘) ↔ 𝑚 = 𝑘))
133127, 132mpbid 221 . . . . . 6 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑚 = 𝑘)
134133ex 449 . . . . 5 (𝜑 → (((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
135134alrimivv 1843 . . . 4 (𝜑 → ∀𝑚𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
136 eleq1 2676 . . . . . 6 (𝑚 = 𝑘 → (𝑚 ∈ ℕ ↔ 𝑘 ∈ ℕ))
137 fveq2 6103 . . . . . . 7 (𝑚 = 𝑘 → (𝑇𝑚) = (𝑇𝑘))
138137eleq2d 2673 . . . . . 6 (𝑚 = 𝑘 → (𝑤 ∈ (𝑇𝑚) ↔ 𝑤 ∈ (𝑇𝑘)))
139136, 138anbi12d 743 . . . . 5 (𝑚 = 𝑘 → ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ↔ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))))
140139mo4 2505 . . . 4 (∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ↔ ∀𝑚𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
141135, 140sylibr 223 . . 3 (𝜑 → ∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
142141alrimiv 1842 . 2 (𝜑 → ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
143 dfdisj2 4555 . 2 (Disj 𝑚 ∈ ℕ (𝑇𝑚) ↔ ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
144142, 143sylibr 223 1 (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ ciun 4455  Disj wdisj 4553   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   Er wer 7626  [cec 7627   / cqs 7628  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   − cmin 10145  -cneg 10146  ℕcn 10897  2c2 10947  ℚcq 11664  [,]cicc 12049  volcvol 23039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-ec 7631  df-qs 7635  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-q 11665  df-icc 12053 This theorem is referenced by:  vitalilem4  23186
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