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Theorem dyadmax 23412
Description: Any nonempty set of dyadic rational intervals has a maximal element. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypothesis
Ref Expression
dyadmbl.1 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
Assertion
Ref Expression
dyadmax ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))
Distinct variable groups:   𝑥,𝑦   𝑧,𝑤,𝑥,𝑦,𝐴   𝑤,𝐹,𝑥,𝑦,𝑧

Proof of Theorem dyadmax
Dummy variables 𝑐 𝑑 𝑎 𝑏 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltweuz 12800 . . . . 5 < We (ℤ‘0)
21a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → < We (ℤ‘0))
3 nn0ex 11336 . . . . . 6 0 ∈ V
43rabex 4845 . . . . 5 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V
54a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V)
6 ssrab2 3720 . . . . . 6 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ ℕ0
7 nn0uz 11760 . . . . . 6 0 = (ℤ‘0)
86, 7sseqtri 3670 . . . . 5 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ (ℤ‘0)
98a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ (ℤ‘0))
10 id 22 . . . . . . 7 (𝐴 ≠ ∅ → 𝐴 ≠ ∅)
11 dyadmbl.1 . . . . . . . . . . . 12 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
1211dyadf 23405 . . . . . . . . . . 11 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
13 ffn 6083 . . . . . . . . . . 11 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14 ovelrn 6852 . . . . . . . . . . 11 (𝐹 Fn (ℤ × ℕ0) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛)))
1512, 13, 14mp2b 10 . . . . . . . . . 10 (𝑧 ∈ ran 𝐹 ↔ ∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛))
16 rexcom 3128 . . . . . . . . . 10 (∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
1715, 16sylbb 209 . . . . . . . . 9 (𝑧 ∈ ran 𝐹 → ∃𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
1817rgen 2951 . . . . . . . 8 𝑧 ∈ ran 𝐹𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)
19 ssralv 3699 . . . . . . . 8 (𝐴 ⊆ ran 𝐹 → (∀𝑧 ∈ ran 𝐹𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) → ∀𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)))
2018, 19mpi 20 . . . . . . 7 (𝐴 ⊆ ran 𝐹 → ∀𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
21 r19.2z 4093 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)) → ∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2210, 20, 21syl2anr 494 . . . . . 6 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
23 rexcom 3128 . . . . . 6 (∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2422, 23sylib 208 . . . . 5 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
25 rabn0 3991 . . . . 5 ({𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅ ↔ ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2624, 25sylibr 224 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)
27 wereu 5139 . . . 4 (( < We (ℤ‘0) ∧ ({𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V ∧ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ (ℤ‘0) ∧ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)) → ∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
282, 5, 9, 26, 27syl13anc 1368 . . 3 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
29 reurex 3190 . . 3 (∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
3028, 29syl 17 . 2 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
31 oveq2 6698 . . . . . . 7 (𝑛 = 𝑐 → (𝑎𝐹𝑛) = (𝑎𝐹𝑐))
3231eqeq2d 2661 . . . . . 6 (𝑛 = 𝑐 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑧 = (𝑎𝐹𝑐)))
33322rexbidv 3086 . . . . 5 (𝑛 = 𝑐 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
3433elrab 3396 . . . 4 (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ↔ (𝑐 ∈ ℕ0 ∧ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
35 eqeq1 2655 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑎𝐹𝑛)))
36 oveq1 6697 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎𝐹𝑛) = (𝑏𝐹𝑛))
3736eqeq2d 2661 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑤 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑛)))
3835, 37cbvrex2v 3210 . . . . . . . . 9 (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛))
39 oveq2 6698 . . . . . . . . . . 11 (𝑛 = 𝑑 → (𝑏𝐹𝑛) = (𝑏𝐹𝑑))
4039eqeq2d 2661 . . . . . . . . . 10 (𝑛 = 𝑑 → (𝑤 = (𝑏𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑑)))
41402rexbidv 3086 . . . . . . . . 9 (𝑛 = 𝑑 → (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
4238, 41syl5bb 272 . . . . . . . 8 (𝑛 = 𝑑 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
4342ralrab 3401 . . . . . . 7 (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 ↔ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
44 r19.23v 3052 . . . . . . . . . . . . . . . . 17 (∀𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
4544ralbii 3009 . . . . . . . . . . . . . . . 16 (∀𝑑 ∈ ℕ0𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
46 ralcom 3127 . . . . . . . . . . . . . . . 16 (∀𝑑 ∈ ℕ0𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
4745, 46bitr3i 266 . . . . . . . . . . . . . . 15 (∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
48 simplll 813 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → 𝐴 ⊆ ran 𝐹)
4948sselda 3636 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → 𝑤 ∈ ran 𝐹)
50 ovelrn 6852 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn (ℤ × ℕ0) → (𝑤 ∈ ran 𝐹 ↔ ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑)))
5112, 13, 50mp2b 10 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ran 𝐹 ↔ ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
5249, 51sylib 208 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
53 rexcom 3128 . . . . . . . . . . . . . . . . . . 19 (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) ↔ ∃𝑑 ∈ ℕ0𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))
54 r19.29 3101 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑑 ∈ ℕ0𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
5554expcom 450 . . . . . . . . . . . . . . . . . . 19 (∃𝑑 ∈ ℕ0𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))))
5653, 55sylbi 207 . . . . . . . . . . . . . . . . . 18 (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))))
5752, 56syl 17 . . . . . . . . . . . . . . . . 17 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))))
58 simplrr 818 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → 𝑎 ∈ ℤ)
5958ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑎 ∈ ℤ)
60 simplrr 818 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑏 ∈ ℤ)
61 simp-5r 826 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑐 ∈ ℕ0)
62 simplrl 817 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑑 ∈ ℕ0)
63 simprl 809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → ¬ 𝑑 < 𝑐)
64 simprr 811 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))
6511, 59, 60, 61, 62, 63, 64dyadmaxlem 23411 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → (𝑎 = 𝑏𝑐 = 𝑑))
66 oveq12 6699 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 = 𝑏𝑐 = 𝑑) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))
6765, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))
6867exp32 630 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
69 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = (𝑏𝐹𝑑) → ([,]‘𝑤) = ([,]‘(𝑏𝐹𝑑)))
7069sseq2d 3666 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑))))
71 eqeq2 2662 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑏𝐹𝑑) → ((𝑎𝐹𝑐) = 𝑤 ↔ (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))
7270, 71imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝑏𝐹𝑑) → ((([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
7372imbi2d 329 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑏𝐹𝑑) → ((¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)) ↔ (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))))
7468, 73syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) → (𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7574anassrs 681 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ 𝑑 ∈ ℕ0) ∧ 𝑏 ∈ ℤ) → (𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7675rexlimdva 3060 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ 𝑑 ∈ ℕ0) → (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7776a2d 29 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ 𝑑 ∈ ℕ0) → ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7877impd 446 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ 𝑑 ∈ ℕ0) → (((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
7978rexlimdva 3060 . . . . . . . . . . . . . . . . 17 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → (∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8057, 79syld 47 . . . . . . . . . . . . . . . 16 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8180ralimdva 2991 . . . . . . . . . . . . . . 15 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → (∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8247, 81syl5bi 232 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → (∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8382imp 444 . . . . . . . . . . . . 13 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))
8483an32s 863 . . . . . . . . . . . 12 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))
85 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑧 = (𝑎𝐹𝑐) → ([,]‘𝑧) = ([,]‘(𝑎𝐹𝑐)))
8685sseq1d 3665 . . . . . . . . . . . . . 14 (𝑧 = (𝑎𝐹𝑐) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤)))
87 eqeq1 2655 . . . . . . . . . . . . . 14 (𝑧 = (𝑎𝐹𝑐) → (𝑧 = 𝑤 ↔ (𝑎𝐹𝑐) = 𝑤))
8886, 87imbi12d 333 . . . . . . . . . . . . 13 (𝑧 = (𝑎𝐹𝑐) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8988ralbidv 3015 . . . . . . . . . . . 12 (𝑧 = (𝑎𝐹𝑐) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
9084, 89syl5ibrcom 237 . . . . . . . . . . 11 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → (𝑧 = (𝑎𝐹𝑐) → ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9190anassrs 681 . . . . . . . . . 10 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ 𝑧𝐴) ∧ 𝑎 ∈ ℤ) → (𝑧 = (𝑎𝐹𝑐) → ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9291rexlimdva 3060 . . . . . . . . 9 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ 𝑧𝐴) → (∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9392reximdva 3046 . . . . . . . 8 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9493ex 449 . . . . . . 7 (((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) → (∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
9543, 94syl5bi 232 . . . . . 6 (((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
9695com23 86 . . . . 5 (((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
9796expimpd 628 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ((𝑐 ∈ ℕ0 ∧ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)) → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
9834, 97syl5bi 232 . . 3 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
9998rexlimdv 3059 . 2 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → (∃𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
10030, 99mpd 15 1 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  ∃!wreu 2943  {crab 2945  Vcvv 3231  cin 3606  wss 3607  c0 3948  cop 4216   class class class wbr 4685   We wwe 5101   × cxp 5141  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113   / cdiv 10722  2c2 11108  0cn0 11330  cz 11415  cuz 11725  [,]cicc 12216  cexp 12900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279
This theorem is referenced by:  dyadmbllem  23413
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