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Theorem dyadmbllem 24127
Description: Lemma for dyadmbl 24128. (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 4834 . . . 4 (𝑎 ([,] “ 𝐴) ↔ ∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖)
2 iccf 12824 . . . . . . 7 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
3 ffn 6507 . . . . . . 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 24119 . . . . . . . . 9 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
8 frn 6513 . . . . . . . . 9 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
97, 8ax-mp 5 . . . . . . . 8 ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
10 inss2 4203 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
11 rexpssxrxp 10674 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
1210, 11sstri 3973 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
139, 12sstri 3973 . . . . . . 7 ran 𝐹 ⊆ (ℝ* × ℝ*)
145, 13sstrdi 3976 . . . . . 6 (𝜑𝐴 ⊆ (ℝ* × ℝ*))
15 eleq2 2898 . . . . . . 7 (𝑖 = ([,]‘𝑡) → (𝑎𝑖𝑎 ∈ ([,]‘𝑡)))
1615rexima 6990 . . . . . 6 (([,] Fn (ℝ* × ℝ*) ∧ 𝐴 ⊆ (ℝ* × ℝ*)) → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖 ↔ ∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡)))
174, 14, 16sylancr 587 . . . . 5 (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖 ↔ ∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡)))
18 ssrab2 4053 . . . . . . . . 9 {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ 𝐴
195adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝐴 ⊆ ran 𝐹)
2018, 19sstrid 3975 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹)
21 simprl 767 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝑡𝐴)
22 ssid 3986 . . . . . . . . . 10 ([,]‘𝑡) ⊆ ([,]‘𝑡)
23 fveq2 6663 . . . . . . . . . . . 12 (𝑎 = 𝑡 → ([,]‘𝑎) = ([,]‘𝑡))
2423sseq2d 3996 . . . . . . . . . . 11 (𝑎 = 𝑡 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑡)))
2524rspcev 3620 . . . . . . . . . 10 ((𝑡𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑡)) → ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
2621, 22, 25sylancl 586 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
27 rabn0 4336 . . . . . . . . 9 ({𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅ ↔ ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
2826, 27sylibr 235 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅)
296dyadmax 24126 . . . . . . . 8 (({𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹 ∧ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅) → ∃𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
3020, 28, 29syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ∃𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
31 fveq2 6663 . . . . . . . . . . 11 (𝑎 = 𝑚 → ([,]‘𝑎) = ([,]‘𝑚))
3231sseq2d 3996 . . . . . . . . . 10 (𝑎 = 𝑚 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
3332elrab 3677 . . . . . . . . 9 (𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
34 simprlr 776 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑡) ⊆ ([,]‘𝑚))
35 simplrr 774 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑡))
3634, 35sseldd 3965 . . . . . . . . . . 11 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑚))
37 simprll 775 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚𝐴)
38 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑤 → ([,]‘𝑎) = ([,]‘𝑤))
3938sseq2d 3996 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑤 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4039elrab 3677 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4140imbi1i 351 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ ((𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
42 impexp 451 . . . . . . . . . . . . . . . . 17 (((𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
4341, 42bitri 276 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
44 impexp 451 . . . . . . . . . . . . . . . . . 18 (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) ↔ (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
45 sstr2 3971 . . . . . . . . . . . . . . . . . . . . 21 (([,]‘𝑡) ⊆ ([,]‘𝑚) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4645ad2antll 725 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4746ancrd 552 . . . . . . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) → (𝑤𝐴 → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
5251ralimdv2 3173 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
5352impr 455 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
54 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑚 → ([,]‘𝑧) = ([,]‘𝑚))
5554sseq1d 3995 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑚 → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘𝑚) ⊆ ([,]‘𝑤)))
56 equequ1 2023 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑚 → (𝑧 = 𝑤𝑚 = 𝑤))
5755, 56imbi12d 346 . . . . . . . . . . . . . . 15 (𝑧 = 𝑚 → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
5857ralbidv 3194 . . . . . . . . . . . . . 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 3680 . . . . . . . . . . . . 13 (𝑚𝐺 ↔ (𝑚𝐴 ∧ ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
6137, 53, 60sylanbrc 583 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚𝐺)
62 ffun 6510 . . . . . . . . . . . . . 14 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
632, 62ax-mp 5 . . . . . . . . . . . . 13 Fun [,]
6459ssrab3 4054 . . . . . . . . . . . . . . . 16 𝐺𝐴
6564, 14sstrid 3975 . . . . . . . . . . . . . . 15 (𝜑𝐺 ⊆ (ℝ* × ℝ*))
662fdmi 6517 . . . . . . . . . . . . . . 15 dom [,] = (ℝ* × ℝ*)
6765, 66sseqtrrdi 4015 . . . . . . . . . . . . . 14 (𝜑𝐺 ⊆ dom [,])
6867ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝐺 ⊆ dom [,])
69 funfvima2 6984 . . . . . . . . . . . . 13 ((Fun [,] ∧ 𝐺 ⊆ dom [,]) → (𝑚𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
7063, 68, 69sylancr 587 . . . . . . . . . . . 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 4835 . . . . . . . . . . 11 ((𝑎 ∈ ([,]‘𝑚) ∧ ([,]‘𝑚) ∈ ([,] “ 𝐺)) → 𝑎 ([,] “ 𝐺))
7336, 71, 72syl2anc 584 . . . . . . . . . 10 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ([,] “ 𝐺))
7473exp32 421 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺))))
7533, 74syl5bi 243 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → (𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺))))
7675rexlimdv 3280 . . . . . . 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 3282 . . . . 5 (𝜑 → (∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡) → 𝑎 ([,] “ 𝐺)))
7917, 78sylbid 241 . . . 4 (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖𝑎 ([,] “ 𝐺)))
801, 79syl5bi 243 . . 3 (𝜑 → (𝑎 ([,] “ 𝐴) → 𝑎 ([,] “ 𝐺)))
8180ssrdv 3970 . 2 (𝜑 ([,] “ 𝐴) ⊆ ([,] “ 𝐺))
82 imass2 5958 . . . 4 (𝐺𝐴 → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8364, 82ax-mp 5 . . 3 ([,] “ 𝐺) ⊆ ([,] “ 𝐴)
84 uniss 4851 . . 3 (([,] “ 𝐺) ⊆ ([,] “ 𝐴) → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8583, 84mp1i 13 . 2 (𝜑 ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8681, 85eqssd 3981 1 (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  cop 4563   cuni 4830   × cxp 5546  dom cdm 5548  ran crn 5549  cima 5551  Fun wfun 6342   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  cr 10524  1c1 10526   + caddc 10528  *cxr 10662  cle 10664   / cdiv 11285  2c2 11680  0cn0 11885  cz 11969  [,]cicc 12729  cexp 13417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-rest 16684  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-top 21430  df-topon 21447  df-bases 21482  df-cmp 21923  df-ovol 23992
This theorem is referenced by:  dyadmbl  24128  mblfinlem1  34810  mblfinlem2  34811
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