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Theorem dyadmbllem 25634
Description: Lemma for dyadmbl 25635. (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 4911 . . . 4 (𝑎 ([,] “ 𝐴) ↔ ∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖)
2 iccf 13488 . . . . . . 7 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
3 ffn 6736 . . . . . . 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 25626 . . . . . . . . 9 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
8 frn 6743 . . . . . . . . 9 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
97, 8ax-mp 5 . . . . . . . 8 ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
10 inss2 4238 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
11 rexpssxrxp 11306 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
1210, 11sstri 3993 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
139, 12sstri 3993 . . . . . . 7 ran 𝐹 ⊆ (ℝ* × ℝ*)
145, 13sstrdi 3996 . . . . . 6 (𝜑𝐴 ⊆ (ℝ* × ℝ*))
15 eleq2 2830 . . . . . . 7 (𝑖 = ([,]‘𝑡) → (𝑎𝑖𝑎 ∈ ([,]‘𝑡)))
1615rexima 7258 . . . . . 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 4080 . . . . . . . . 9 {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ 𝐴
195adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝐴 ⊆ ran 𝐹)
2018, 19sstrid 3995 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹)
21 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → 𝑡𝐴)
22 ssid 4006 . . . . . . . . . 10 ([,]‘𝑡) ⊆ ([,]‘𝑡)
23 fveq2 6906 . . . . . . . . . . . 12 (𝑎 = 𝑡 → ([,]‘𝑎) = ([,]‘𝑡))
2423sseq2d 4016 . . . . . . . . . . 11 (𝑎 = 𝑡 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑡)))
2524rspcev 3622 . . . . . . . . . 10 ((𝑡𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑡)) → ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
2621, 22, 25sylancl 586 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
27 rabn0 4389 . . . . . . . . 9 ({𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅ ↔ ∃𝑎𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
2826, 27sylibr 234 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅)
296dyadmax 25633 . . . . . . . 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 6906 . . . . . . . . . . 11 (𝑎 = 𝑚 → ([,]‘𝑎) = ([,]‘𝑚))
3231sseq2d 4016 . . . . . . . . . 10 (𝑎 = 𝑚 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
3332elrab 3692 . . . . . . . . 9 (𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
34 simprlr 780 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑡) ⊆ ([,]‘𝑚))
35 simplrr 778 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑡))
3634, 35sseldd 3984 . . . . . . . . . . 11 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑚))
37 simprll 779 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚𝐴)
38 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑤 → ([,]‘𝑎) = ([,]‘𝑤))
3938sseq2d 4016 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑤 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4039elrab 3692 . . . . . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . . . . . . . . 21 (([,]‘𝑡) ⊆ ([,]‘𝑚) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
4645ad2antll 729 . . . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . . 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 242 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) → (𝑤𝐴 → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
5251ralimdv2 3163 . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑚 → ([,]‘𝑧) = ([,]‘𝑚))
5554sseq1d 4015 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑚 → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘𝑚) ⊆ ([,]‘𝑤)))
56 equequ1 2024 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑚 → (𝑧 = 𝑤𝑚 = 𝑤))
5755, 56imbi12d 344 . . . . . . . . . . . . . . 15 (𝑧 = 𝑚 → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
5857ralbidv 3178 . . . . . . . . . . . . . 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 3695 . . . . . . . . . . . . 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 6739 . . . . . . . . . . . . . 14 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
632, 62ax-mp 5 . . . . . . . . . . . . 13 Fun [,]
6459ssrab3 4082 . . . . . . . . . . . . . . . 16 𝐺𝐴
6564, 14sstrid 3995 . . . . . . . . . . . . . . 15 (𝜑𝐺 ⊆ (ℝ* × ℝ*))
662fdmi 6747 . . . . . . . . . . . . . . 15 dom [,] = (ℝ* × ℝ*)
6765, 66sseqtrrdi 4025 . . . . . . . . . . . . . 14 (𝜑𝐺 ⊆ dom [,])
6867ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝐺 ⊆ dom [,])
69 funfvima2 7251 . . . . . . . . . . . . 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 4912 . . . . . . . . . . 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 420 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → ((𝑚𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺))))
7533, 74biimtrid 242 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐴𝑎 ∈ ([,]‘𝑡))) → (𝑚 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (∀𝑤 ∈ {𝑎𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ([,] “ 𝐺))))
7675rexlimdv 3153 . . . . . . 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 3156 . . . . 5 (𝜑 → (∃𝑡𝐴 𝑎 ∈ ([,]‘𝑡) → 𝑎 ([,] “ 𝐺)))
7917, 78sylbid 240 . . . 4 (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎𝑖𝑎 ([,] “ 𝐺)))
801, 79biimtrid 242 . . 3 (𝜑 → (𝑎 ([,] “ 𝐴) → 𝑎 ([,] “ 𝐺)))
8180ssrdv 3989 . 2 (𝜑 ([,] “ 𝐴) ⊆ ([,] “ 𝐺))
82 imass2 6120 . . . 4 (𝐺𝐴 → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8364, 82ax-mp 5 . . 3 ([,] “ 𝐺) ⊆ ([,] “ 𝐴)
84 uniss 4915 . . 3 (([,] “ 𝐺) ⊆ ([,] “ 𝐴) → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8583, 84mp1i 13 . 2 (𝜑 ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
8681, 85eqssd 4001 1 (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  cop 4632   cuni 4907   × cxp 5683  dom cdm 5685  ran crn 5686  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  cr 11154  1c1 11156   + caddc 11158  *cxr 11294  cle 11296   / cdiv 11920  2c2 12321  0cn0 12526  cz 12613  [,]cicc 13390  cexp 14102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499
This theorem is referenced by:  dyadmbl  25635  mblfinlem1  37664  mblfinlem2  37665
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