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Theorem dyadmbllem 23766
Description: Lemma for dyadmbl 23767. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypotheses
Ref Expression
dyadmbl.1 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
dyadmbl.2 𝐺 = {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}
dyadmbl.3 (𝜑𝐴 ⊆ ran 𝐹)
Assertion
Ref Expression
dyadmbllem (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
Distinct variable groups:   𝑥,𝑦   𝑧,𝑤,𝜑   𝑥,𝑤,𝑦,𝐴,𝑧   𝑧,𝐺   𝑤,𝐹,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐺(𝑥,𝑦,𝑤)

Proof of Theorem dyadmbllem
Dummy variables 𝑎 𝑚 𝑡 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 4663 . . . 4 (𝑎 ([,] “ 𝐴) ↔ ∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖)
2 iccf 12562 . . . . . . 7 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
3 ffn 6279 . . . . . . 7 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
42, 3ax-mp 5 . . . . . 6 [,] Fn (ℝ* × ℝ*)
5 dyadmbl.3 . . . . . . 7 (𝜑𝐴 ⊆ ran 𝐹)
6 dyadmbl.1 . . . . . . . . . 10 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
76dyadf 23758 . . . . . . . . 9 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
8 frn 6285 . . . . . . . . 9 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
97, 8ax-mp 5 . . . . . . . 8 ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
10 inss2 4059 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
11 rexpssxrxp 10402 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
1210, 11sstri 3837 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
139, 12sstri 3837 . . . . . . 7 ran 𝐹 ⊆ (ℝ* × ℝ*)
145, 13syl6ss 3840 . . . . . 6 (𝜑𝐴 ⊆ (ℝ* × ℝ*))
15 eleq2 2896 . . . . . . 7 (𝑖 = ([,]‘𝑡) → (𝑎𝑖𝑎 ∈ ([,]‘𝑡)))
1615rexima 6754 . . . . . 6 (([,] Fn (ℝ* × ℝ*) ∧ 𝐴 ⊆ (ℝ* × ℝ*)) → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖 ↔ ∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡)))
174, 14, 16sylancr 583 . . . . 5 (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖 ↔ ∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡)))
18 ssrab2 3913 . . . . . . . . 9 {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ 𝐴
195adantr 474 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝐴 ⊆ ran 𝐹)
2018, 19syl5ss 3839 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹)
21 simprl 789 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝑡𝐴)
22 ssid 3849 . . . . . . . . . 10 ([,]‘𝑡) ⊆ ([,]‘𝑡)
23 fveq2 6434 . . . . . . . . . . . 12 (𝑎 = 𝑡 → ([,]‘𝑎) = ([,]‘𝑡))
2423sseq2d 3859 . . . . . . . . . . 11 (𝑎 = 𝑡 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑡)))
2524rspcev 3527 . . . . . . . . . 10 ((𝑡𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑡)) → ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
2621, 22, 25sylancl 582 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
27 rabn0 4188 . . . . . . . . 9 ({𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅ ↔ ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
2826, 27sylibr 226 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅)
296dyadmax 23765 . . . . . . . 8 (({𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹 ∧ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅) → ∃𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
3020, 28, 29syl2anc 581 . . . . . . 7 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ∃𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
31 fveq2 6434 . . . . . . . . . . 11 (𝑎 = 𝑚 → ([,]‘𝑎) = ([,]‘𝑚))
3231sseq2d 3859 . . . . . . . . . 10 (𝑎 = 𝑚 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
3332elrab 3586 . . . . . . . . 9 (𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
34 simprlr 800 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑡) ⊆ ([,]‘𝑚))
35 simplrr 798 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑡))
3634, 35sseldd 3829 . . . . . . . . . . 11 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑚))
37 simprll 799 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚𝐴)
38 fveq2 6434 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑤 → ([,]‘𝑎) = ([,]‘𝑤))
3938sseq2d 3859 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑤 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4039elrab 3586 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4140imbi1i 341 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ ((𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
42 impexp 443 . . . . . . . . . . . . . . . . 17 (((𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
4341, 42bitri 267 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
44 impexp 443 . . . . . . . . . . . . . . . . . 18 (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) ↔ (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
45 sstr2 3835 . . . . . . . . . . . . . . . . . . . . 21 (([,]‘𝑡) ⊆ ([,]‘𝑚) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4645ad2antll 722 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4746ancrd 549 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → (([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤))))
4847imim1d 82 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
4944, 48syl5bir 235 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
5049imim2d 57 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((𝑤𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → (𝑤𝐴 → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
5143, 50syl5bi 234 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) → (𝑤𝐴 → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
5251ralimdv2 3171 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
5352impr 448 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
54 fveq2 6434 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑚 → ([,]‘𝑧) = ([,]‘𝑚))
5554sseq1d 3858 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑚 → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘𝑚) ⊆ ([,]‘𝑤)))
56 equequ1 2131 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑚 → (𝑧 = 𝑤𝑚 = 𝑤))
5755, 56imbi12d 336 . . . . . . . . . . . . . . 15 (𝑧 = 𝑚 → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
5857ralbidv 3196 . . . . . . . . . . . . . 14 (𝑧 = 𝑚 → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
59 dyadmbl.2 . . . . . . . . . . . . . 14 𝐺 = {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}
6058, 59elrab2 3590 . . . . . . . . . . . . 13 (𝑚𝐺 ↔ (𝑚𝐴 ∧ ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
6137, 53, 60sylanbrc 580 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚𝐺)
62 ffun 6282 . . . . . . . . . . . . . 14 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
632, 62ax-mp 5 . . . . . . . . . . . . 13 Fun [,]
64 ssrab2 3913 . . . . . . . . . . . . . . . . 17 {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)} ⊆ 𝐴
6559, 64eqsstri 3861 . . . . . . . . . . . . . . . 16 𝐺𝐴
6665, 14syl5ss 3839 . . . . . . . . . . . . . . 15 (𝜑𝐺 ⊆ (ℝ* × ℝ*))
672fdmi 6289 . . . . . . . . . . . . . . 15 dom [,] = (ℝ* × ℝ*)
6866, 67syl6sseqr 3878 . . . . . . . . . . . . . 14 (𝜑𝐺 ⊆ dom [,])
6968ad2antrr 719 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝐺 ⊆ dom [,])
70 funfvima2 6750 . . . . . . . . . . . . 13 ((Fun [,] ∧ 𝐺 ⊆ dom [,]) → (𝑚𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
7163, 69, 70sylancr 583 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → (𝑚𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
7261, 71mpd 15 . . . . . . . . . . 11 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑚) ∈ ([,] “ 𝐺))
73 elunii 4664 . . . . . . . . . . 11 ((𝑎 ∈ ([,]‘𝑚) ∧ ([,]‘𝑚) ∈ ([,] “ 𝐺)) → 𝑎 ([,] “ 𝐺))
7436, 72, 73syl2anc 581 . . . . . . . . . 10 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ([,] “ 𝐺))
7574exp32 413 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺))))
7633, 75syl5bi 234 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → (𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺))))
7776rexlimdv 3240 . . . . . . 7 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → (∃𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺)))
7830, 77mpd 15 . . . . . 6 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝑎 ([,] “ 𝐺))
7978rexlimdvaa 3242 . . . . 5 (𝜑 → (∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡) → 𝑎 ([,] “ 𝐺)))
8017, 79sylbid 232 . . . 4 (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖𝑎 ([,] “ 𝐺)))
811, 80syl5bi 234 . . 3 (𝜑 → (𝑎 ([,] “ 𝐴) → 𝑎 ([,] “ 𝐺)))
8281ssrdv 3834 . 2 (𝜑 ([,] “ 𝐴) ⊆ ([,] “ 𝐺))
83 imass2 5743 . . . 4 (𝐺𝐴 → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8465, 83ax-mp 5 . . 3 ([,] “ 𝐺) ⊆ ([,] “ 𝐴)
85 uniss 4682 . . 3 (([,] “ 𝐺) ⊆ ([,] “ 𝐴) → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8684, 85mp1i 13 . 2 (𝜑 ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8782, 86eqssd 3845 1 (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wne 3000  wral 3118  wrex 3119  {crab 3122  cin 3798  wss 3799  c0 4145  𝒫 cpw 4379  cop 4404   cuni 4659   × cxp 5341  dom cdm 5343  ran crn 5344  cima 5346  Fun wfun 6118   Fn wfn 6119  wf 6120  cfv 6124  (class class class)co 6906  cmpt2 6908  cr 10252  1c1 10254   + caddc 10256  *cxr 10391  cle 10393   / cdiv 11010  2c2 11407  0cn0 11619  cz 11705  [,]cicc 12467  cexp 13155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-inf2 8816  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330  ax-pre-sup 10331
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-se 5303  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-isom 6133  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-map 8125  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-fi 8587  df-sup 8618  df-inf 8619  df-oi 8685  df-card 9079  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-div 11011  df-nn 11352  df-2 11415  df-3 11416  df-n0 11620  df-z 11706  df-uz 11970  df-q 12073  df-rp 12114  df-xneg 12233  df-xadd 12234  df-xmul 12235  df-ioo 12468  df-ico 12470  df-icc 12471  df-fz 12621  df-fzo 12762  df-seq 13097  df-exp 13156  df-hash 13412  df-cj 14217  df-re 14218  df-im 14219  df-sqrt 14353  df-abs 14354  df-clim 14597  df-sum 14795  df-rest 16437  df-topgen 16458  df-psmet 20099  df-xmet 20100  df-met 20101  df-bl 20102  df-mopn 20103  df-top 21070  df-topon 21087  df-bases 21122  df-cmp 21562  df-ovol 23631
This theorem is referenced by:  dyadmbl  23767  mblfinlem1  33991  mblfinlem2  33992
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