MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dyadmbllem Structured version   Visualization version   GIF version

Theorem dyadmbllem 24206
Description: Lemma for dyadmbl 24207. (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 4807 . . . 4 (𝑎 ([,] “ 𝐴) ↔ ∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖)
2 iccf 12830 . . . . . . 7 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
3 ffn 6491 . . . . . . 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 24198 . . . . . . . . 9 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
8 frn 6497 . . . . . . . . 9 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
97, 8ax-mp 5 . . . . . . . 8 ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
10 inss2 4159 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
11 rexpssxrxp 10679 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
1210, 11sstri 3927 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
139, 12sstri 3927 . . . . . . 7 ran 𝐹 ⊆ (ℝ* × ℝ*)
145, 13sstrdi 3930 . . . . . 6 (𝜑𝐴 ⊆ (ℝ* × ℝ*))
15 eleq2 2881 . . . . . . 7 (𝑖 = ([,]‘𝑡) → (𝑎𝑖𝑎 ∈ ([,]‘𝑡)))
1615rexima 6981 . . . . . 6 (([,] Fn (ℝ* × ℝ*) ∧ 𝐴 ⊆ (ℝ* × ℝ*)) → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖 ↔ ∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡)))
174, 14, 16sylancr 590 . . . . 5 (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖 ↔ ∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡)))
18 ssrab2 4010 . . . . . . . . 9 {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ 𝐴
195adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝐴 ⊆ ran 𝐹)
2018, 19sstrid 3929 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹)
21 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝑡𝐴)
22 ssid 3940 . . . . . . . . . 10 ([,]‘𝑡) ⊆ ([,]‘𝑡)
23 fveq2 6649 . . . . . . . . . . . 12 (𝑎 = 𝑡 → ([,]‘𝑎) = ([,]‘𝑡))
2423sseq2d 3950 . . . . . . . . . . 11 (𝑎 = 𝑡 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑡)))
2524rspcev 3574 . . . . . . . . . 10 ((𝑡𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑡)) → ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
2621, 22, 25sylancl 589 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
27 rabn0 4296 . . . . . . . . 9 ({𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅ ↔ ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
2826, 27sylibr 237 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅)
296dyadmax 24205 . . . . . . . 8 (({𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹 ∧ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅) → ∃𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
3020, 28, 29syl2anc 587 . . . . . . 7 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ∃𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
31 fveq2 6649 . . . . . . . . . . 11 (𝑎 = 𝑚 → ([,]‘𝑎) = ([,]‘𝑚))
3231sseq2d 3950 . . . . . . . . . 10 (𝑎 = 𝑚 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
3332elrab 3631 . . . . . . . . 9 (𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
34 simprlr 779 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑡) ⊆ ([,]‘𝑚))
35 simplrr 777 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑡))
3634, 35sseldd 3919 . . . . . . . . . . 11 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑚))
37 simprll 778 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚𝐴)
38 fveq2 6649 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑤 → ([,]‘𝑎) = ([,]‘𝑤))
3938sseq2d 3950 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑤 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4039elrab 3631 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4140imbi1i 353 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ ((𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
42 impexp 454 . . . . . . . . . . . . . . . . 17 (((𝑤𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
4341, 42bitri 278 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
44 impexp 454 . . . . . . . . . . . . . . . . . 18 (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) ↔ (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
45 sstr2 3925 . . . . . . . . . . . . . . . . . . . . 21 (([,]‘𝑡) ⊆ ([,]‘𝑚) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4645ad2antll 728 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4746ancrd 555 . . . . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) → (𝑤𝐴 → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
5251ralimdv2 3146 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
5352impr 458 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
54 fveq2 6649 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑚 → ([,]‘𝑧) = ([,]‘𝑚))
5554sseq1d 3949 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑚 → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘𝑚) ⊆ ([,]‘𝑤)))
56 equequ1 2032 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑚 → (𝑧 = 𝑤𝑚 = 𝑤))
5755, 56imbi12d 348 . . . . . . . . . . . . . . 15 (𝑧 = 𝑚 → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
5857ralbidv 3165 . . . . . . . . . . . . . 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 3634 . . . . . . . . . . . . 13 (𝑚𝐺 ↔ (𝑚𝐴 ∧ ∀𝑤𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
6137, 53, 60sylanbrc 586 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚𝐺)
62 ffun 6494 . . . . . . . . . . . . . 14 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
632, 62ax-mp 5 . . . . . . . . . . . . 13 Fun [,]
6459ssrab3 4011 . . . . . . . . . . . . . . . 16 𝐺𝐴
6564, 14sstrid 3929 . . . . . . . . . . . . . . 15 (𝜑𝐺 ⊆ (ℝ* × ℝ*))
662fdmi 6502 . . . . . . . . . . . . . . 15 dom [,] = (ℝ* × ℝ*)
6765, 66sseqtrrdi 3969 . . . . . . . . . . . . . 14 (𝜑𝐺 ⊆ dom [,])
6867ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝐺 ⊆ dom [,])
69 funfvima2 6975 . . . . . . . . . . . . 13 ((Fun [,] ∧ 𝐺 ⊆ dom [,]) → (𝑚𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
7063, 68, 69sylancr 590 . . . . . . . . . . . 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 4808 . . . . . . . . . . 11 ((𝑎 ∈ ([,]‘𝑚) ∧ ([,]‘𝑚) ∈ ([,] “ 𝐺)) → 𝑎 ([,] “ 𝐺))
7336, 71, 72syl2anc 587 . . . . . . . . . 10 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ([,] “ 𝐺))
7473exp32 424 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺))))
7533, 74syl5bi 245 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → (𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺))))
7675rexlimdv 3245 . . . . . . 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 3247 . . . . 5 (𝜑 → (∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡) → 𝑎 ([,] “ 𝐺)))
7917, 78sylbid 243 . . . 4 (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖𝑎 ([,] “ 𝐺)))
801, 79syl5bi 245 . . 3 (𝜑 → (𝑎 ([,] “ 𝐴) → 𝑎 ([,] “ 𝐺)))
8180ssrdv 3924 . 2 (𝜑 ([,] “ 𝐴) ⊆ ([,] “ 𝐺))
82 imass2 5936 . . . 4 (𝐺𝐴 → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8364, 82ax-mp 5 . . 3 ([,] “ 𝐺) ⊆ ([,] “ 𝐴)
84 uniss 4811 . . 3 (([,] “ 𝐺) ⊆ ([,] “ 𝐴) → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8583, 84mp1i 13 . 2 (𝜑 ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8681, 85eqssd 3935 1 (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  {crab 3113  cin 3883  wss 3884  c0 4246  𝒫 cpw 4500  cop 4534   cuni 4803   × cxp 5521  dom cdm 5523  ran crn 5524  cima 5526  Fun wfun 6322   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  cr 10529  1c1 10531   + caddc 10533  *cxr 10667  cle 10669   / cdiv 11290  2c2 11684  0cn0 11889  cz 11973  [,]cicc 12733  cexp 13429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-sum 15038  df-rest 16691  df-topgen 16712  df-psmet 20086  df-xmet 20087  df-met 20088  df-bl 20089  df-mopn 20090  df-top 21502  df-topon 21519  df-bases 21554  df-cmp 21995  df-ovol 24071
This theorem is referenced by:  dyadmbl  24207  mblfinlem1  35087  mblfinlem2  35088
  Copyright terms: Public domain W3C validator