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Theorem dyadmbllem 25447
Description: Lemma for dyadmbl 25448. (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 4912 . . . 4 (𝑎 ([,] “ 𝐴) ↔ ∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖)
2 iccf 13432 . . . . . . 7 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
3 ffn 6717 . . . . . . 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 25439 . . . . . . . . 9 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
8 frn 6724 . . . . . . . . 9 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
97, 8ax-mp 5 . . . . . . . 8 ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
10 inss2 4229 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
11 rexpssxrxp 11266 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
1210, 11sstri 3991 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
139, 12sstri 3991 . . . . . . 7 ran 𝐹 ⊆ (ℝ* × ℝ*)
145, 13sstrdi 3994 . . . . . 6 (𝜑𝐴 ⊆ (ℝ* × ℝ*))
15 eleq2 2821 . . . . . . 7 (𝑖 = ([,]‘𝑡) → (𝑎𝑖𝑎 ∈ ([,]‘𝑡)))
1615rexima 7241 . . . . . 6 (([,] Fn (ℝ* × ℝ*) ∧ 𝐴 ⊆ (ℝ* × ℝ*)) → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖 ↔ ∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡)))
174, 14, 16sylancr 586 . . . . 5 (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖 ↔ ∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡)))
18 ssrab2 4077 . . . . . . . . 9 {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ 𝐴
195adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝐴 ⊆ ran 𝐹)
2018, 19sstrid 3993 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹)
21 simprl 768 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝑡𝐴)
22 ssid 4004 . . . . . . . . . 10 ([,]‘𝑡) ⊆ ([,]‘𝑡)
23 fveq2 6891 . . . . . . . . . . . 12 (𝑎 = 𝑡 → ([,]‘𝑎) = ([,]‘𝑡))
2423sseq2d 4014 . . . . . . . . . . 11 (𝑎 = 𝑡 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑡)))
2524rspcev 3612 . . . . . . . . . 10 ((𝑡𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑡)) → ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
2621, 22, 25sylancl 585 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
27 rabn0 4385 . . . . . . . . 9 ({𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅ ↔ ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
2826, 27sylibr 233 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅)
296dyadmax 25446 . . . . . . . 8 (({𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹 ∧ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅) → ∃𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
3020, 28, 29syl2anc 583 . . . . . . 7 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ∃𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
31 fveq2 6891 . . . . . . . . . . 11 (𝑎 = 𝑚 → ([,]‘𝑎) = ([,]‘𝑚))
3231sseq2d 4014 . . . . . . . . . 10 (𝑎 = 𝑚 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
3332elrab 3683 . . . . . . . . 9 (𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
34 simprlr 777 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑡) ⊆ ([,]‘𝑚))
35 simplrr 775 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑡))
3634, 35sseldd 3983 . . . . . . . . . . 11 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑚))
37 simprll 776 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚𝐴)
38 fveq2 6891 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑤 → ([,]‘𝑎) = ([,]‘𝑤))
3938sseq2d 4014 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑤 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4039elrab 3683 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4140imbi1i 349 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ ((𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
42 impexp 450 . . . . . . . . . . . . . . . . 17 (((𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
4341, 42bitri 275 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
44 impexp 450 . . . . . . . . . . . . . . . . . 18 (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) ↔ (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
45 sstr2 3989 . . . . . . . . . . . . . . . . . . . . 21 (([,]‘𝑡) ⊆ ([,]‘𝑚) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4645ad2antll 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4746ancrd 551 . . . . . . . . . . . . . . . . . . 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, 48biimtrrid 242 . . . . . . . . . . . . . . . . 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, 50biimtrid 241 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) → (𝑤𝐴 → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
5251ralimdv2 3162 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
5352impr 454 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
54 fveq2 6891 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑚 → ([,]‘𝑧) = ([,]‘𝑚))
5554sseq1d 4013 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑚 → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘𝑚) ⊆ ([,]‘𝑤)))
56 equequ1 2027 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑚 → (𝑧 = 𝑤𝑚 = 𝑤))
5755, 56imbi12d 344 . . . . . . . . . . . . . . 15 (𝑧 = 𝑚 → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
5857ralbidv 3176 . . . . . . . . . . . . . 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 3686 . . . . . . . . . . . . 13 (𝑚𝐺 ↔ (𝑚𝐴 ∧ ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
6137, 53, 60sylanbrc 582 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚𝐺)
62 ffun 6720 . . . . . . . . . . . . . 14 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
632, 62ax-mp 5 . . . . . . . . . . . . 13 Fun [,]
6459ssrab3 4080 . . . . . . . . . . . . . . . 16 𝐺𝐴
6564, 14sstrid 3993 . . . . . . . . . . . . . . 15 (𝜑𝐺 ⊆ (ℝ* × ℝ*))
662fdmi 6729 . . . . . . . . . . . . . . 15 dom [,] = (ℝ* × ℝ*)
6765, 66sseqtrrdi 4033 . . . . . . . . . . . . . 14 (𝜑𝐺 ⊆ dom [,])
6867ad2antrr 723 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝐺 ⊆ dom [,])
69 funfvima2 7235 . . . . . . . . . . . . 13 ((Fun [,] ∧ 𝐺 ⊆ dom [,]) → (𝑚𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
7063, 68, 69sylancr 586 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → (𝑚𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
7161, 70mpd 15 . . . . . . . . . . 11 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑚) ∈ ([,] “ 𝐺))
72 elunii 4913 . . . . . . . . . . 11 ((𝑎 ∈ ([,]‘𝑚) ∧ ([,]‘𝑚) ∈ ([,] “ 𝐺)) → 𝑎 ([,] “ 𝐺))
7336, 71, 72syl2anc 583 . . . . . . . . . 10 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ([,] “ 𝐺))
7473exp32 420 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺))))
7533, 74biimtrid 241 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → (𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺))))
7675rexlimdv 3152 . . . . . . 7 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → (∃𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺)))
7730, 76mpd 15 . . . . . 6 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝑎 ([,] “ 𝐺))
7877rexlimdvaa 3155 . . . . 5 (𝜑 → (∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡) → 𝑎 ([,] “ 𝐺)))
7917, 78sylbid 239 . . . 4 (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖𝑎 ([,] “ 𝐺)))
801, 79biimtrid 241 . . 3 (𝜑 → (𝑎 ([,] “ 𝐴) → 𝑎 ([,] “ 𝐺)))
8180ssrdv 3988 . 2 (𝜑 ([,] “ 𝐴) ⊆ ([,] “ 𝐺))
82 imass2 6101 . . . 4 (𝐺𝐴 → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8364, 82ax-mp 5 . . 3 ([,] “ 𝐺) ⊆ ([,] “ 𝐴)
84 uniss 4916 . . 3 (([,] “ 𝐺) ⊆ ([,] “ 𝐴) → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8583, 84mp1i 13 . 2 (𝜑 ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8681, 85eqssd 3999 1 (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wne 2939  wral 3060  wrex 3069  {crab 3431  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602  cop 4634   cuni 4908   × cxp 5674  dom cdm 5676  ran crn 5677  cima 5679  Fun wfun 6537   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  cmpo 7414  cr 11115  1c1 11117   + caddc 11119  *cxr 11254  cle 11256   / cdiv 11878  2c2 12274  0cn0 12479  cz 12565  [,]cicc 13334  cexp 14034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fi 9412  df-sup 9443  df-inf 9444  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-q 12940  df-rp 12982  df-xneg 13099  df-xadd 13100  df-xmul 13101  df-ioo 13335  df-ico 13337  df-icc 13338  df-fz 13492  df-fzo 13635  df-seq 13974  df-exp 14035  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-sum 15640  df-rest 17375  df-topgen 17396  df-psmet 21224  df-xmet 21225  df-met 21226  df-bl 21227  df-mopn 21228  df-top 22715  df-topon 22732  df-bases 22768  df-cmp 23210  df-ovol 25312
This theorem is referenced by:  dyadmbl  25448  mblfinlem1  36988  mblfinlem2  36989
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