Theorem mblfinlem2

Description: Lemma for ismblfin , effectively one direction of the same fact for open sets, made necessary by Viaclovsky's slightly different definition of outer measure. Note that unlike the main theorem, this holds for sets of infinite measure. (Contributed by Brendan Leahy, 21-Feb-2018) (Revised by Brendan Leahy, 13-Jul-2018)

	Ref	Expression
Assertion	mblfinlem2	⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )

Proof

Step	Hyp	Ref	Expression
1		retop	⊢ ( topGen ‘ ran (,) ) ∈ Top
2		0cld	⊢ ( ( topGen ‘ ran (,) ) ∈ Top → ∅ ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
3	1 2	ax-mp	⊢ ∅ ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) )
4		simpl3	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → 𝑀 < ( vol* ‘ 𝐴 ) )
5		fveq2	⊢ ( 𝐴 = ∅ → ( vol* ‘ 𝐴 ) = ( vol* ‘ ∅ ) )
6	5	adantl	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → ( vol* ‘ 𝐴 ) = ( vol* ‘ ∅ ) )
7	4 6	breqtrd	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → 𝑀 < ( vol* ‘ ∅ ) )
8		0ss	⊢ ∅ ⊆ 𝐴
9	7 8	jctil	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → ( ∅ ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ ∅ ) ) )
10		sseq1	⊢ ( 𝑠 = ∅ → ( 𝑠 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴 ) )
11		fveq2	⊢ ( 𝑠 = ∅ → ( vol* ‘ 𝑠 ) = ( vol* ‘ ∅ ) )
12	11	breq2d	⊢ ( 𝑠 = ∅ → ( 𝑀 < ( vol* ‘ 𝑠 ) ↔ 𝑀 < ( vol* ‘ ∅ ) ) )
13	10 12	anbi12d	⊢ ( 𝑠 = ∅ → ( ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) ↔ ( ∅ ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ ∅ ) ) ) )
14	13	rspcev	⊢ ( ( ∅ ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ∧ ( ∅ ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ ∅ ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )
15	3 9 14	sylancr	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )
16		mblfinlem1	⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝐴 ≠ ∅ ) → ∃ 𝑓 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } )
17	16	3ad2antl1	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 ≠ ∅ ) → ∃ 𝑓 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } )
18		simpl3	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → 𝑀 < ( vol* ‘ 𝐴 ) )
19		f1ofo	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → 𝑓 : ℕ –onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } )
20		rnco2	⊢ ran ( [,] ∘ 𝑓 ) = ( [,] “ ran 𝑓 )
21		forn	⊢ ( 𝑓 : ℕ –onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ran 𝑓 = { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } )
22	21	imaeq2d	⊢ ( 𝑓 : ℕ –onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( [,] “ ran 𝑓 ) = ( [,] “ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) )
23	20 22	eqtrid	⊢ ( 𝑓 : ℕ –onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ran ( [,] ∘ 𝑓 ) = ( [,] “ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) )
24	23	unieqd	⊢ ( 𝑓 : ℕ –onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∪ ran ( [,] ∘ 𝑓 ) = ∪ ( [,] “ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) )
25	19 24	syl	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∪ ran ( [,] ∘ 𝑓 ) = ∪ ( [,] “ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) )
26	25	adantl	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ∪ ran ( [,] ∘ 𝑓 ) = ∪ ( [,] “ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) )
27		oveq1	⊢ ( 𝑥 = 𝑢 → ( 𝑥 / ( 2 ↑ 𝑦 ) ) = ( 𝑢 / ( 2 ↑ 𝑦 ) ) )
28		oveq1	⊢ ( 𝑥 = 𝑢 → ( 𝑥 + 1 ) = ( 𝑢 + 1 ) )
29	28	oveq1d	⊢ ( 𝑥 = 𝑢 → ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) = ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑦 ) ) )
30	27 29	opeq12d	⊢ ( 𝑥 = 𝑢 → ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ = ⟨ ( 𝑢 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ )
31		oveq2	⊢ ( 𝑦 = 𝑣 → ( 2 ↑ 𝑦 ) = ( 2 ↑ 𝑣 ) )
32	31	oveq2d	⊢ ( 𝑦 = 𝑣 → ( 𝑢 / ( 2 ↑ 𝑦 ) ) = ( 𝑢 / ( 2 ↑ 𝑣 ) ) )
33	31	oveq2d	⊢ ( 𝑦 = 𝑣 → ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑦 ) ) = ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑣 ) ) )
34	32 33	opeq12d	⊢ ( 𝑦 = 𝑣 → ⟨ ( 𝑢 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ = ⟨ ( 𝑢 / ( 2 ↑ 𝑣 ) ) , ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑣 ) ) ⟩ )
35	30 34	cbvmpov	⊢ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) = ( 𝑢 ∈ ℤ , 𝑣 ∈ ℕ0 ↦ ⟨ ( 𝑢 / ( 2 ↑ 𝑣 ) ) , ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑣 ) ) ⟩ )
36		fveq2	⊢ ( 𝑎 = 𝑧 → ( [,] ‘ 𝑎 ) = ( [,] ‘ 𝑧 ) )
37	36	sseq1d	⊢ ( 𝑎 = 𝑧 → ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ 𝑧 ) ⊆ ( [,] ‘ 𝑐 ) ) )
38		eqeq1	⊢ ( 𝑎 = 𝑧 → ( 𝑎 = 𝑐 ↔ 𝑧 = 𝑐 ) )
39	37 38	imbi12d	⊢ ( 𝑎 = 𝑧 → ( ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ( ( [,] ‘ 𝑧 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑧 = 𝑐 ) ) )
40	39	ralbidv	⊢ ( 𝑎 = 𝑧 → ( ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑧 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑧 = 𝑐 ) ) )
41	40	cbvrabv	⊢ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } = { 𝑧 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑧 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑧 = 𝑐 ) }
42		ssrab2	⊢ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ⊆ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ )
43	42	a1i	⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) → { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ⊆ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) )
44	35 41 43	dyadmbllem	⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) → ∪ ( [,] “ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ) = ∪ ( [,] “ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) )
45	44	adantr	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ∪ ( [,] “ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ) = ∪ ( [,] “ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) )
46	26 45	eqtr4d	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ∪ ran ( [,] ∘ 𝑓 ) = ∪ ( [,] “ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ) )
47		opnmbllem0	⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ∪ ( [,] “ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ) = 𝐴 )
48	47	3ad2ant1	⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) → ∪ ( [,] “ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ) = 𝐴 )
49	48	adantr	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ∪ ( [,] “ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ) = 𝐴 )
50	46 49	eqtrd	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ∪ ran ( [,] ∘ 𝑓 ) = 𝐴 )
51	50	fveq2d	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( vol* ‘ ∪ ran ( [,] ∘ 𝑓 ) ) = ( vol* ‘ 𝐴 ) )
52		f1of	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } )
53		ssrab2	⊢ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ⊆ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 }
54	35	dyadf	⊢ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) )
55		frn	⊢ ( ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) → ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ⊆ ( ≤ ∩ ( ℝ × ℝ ) ) )
56	54 55	ax-mp	⊢ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ⊆ ( ≤ ∩ ( ℝ × ℝ ) )
57	42 56	sstri	⊢ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ⊆ ( ≤ ∩ ( ℝ × ℝ ) )
58	53 57	sstri	⊢ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ⊆ ( ≤ ∩ ( ℝ × ℝ ) )
59		fss	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ⊆ ( ≤ ∩ ( ℝ × ℝ ) ) ) → 𝑓 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) )
60	52 58 59	sylancl	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → 𝑓 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) )
61	53 42	sstri	⊢ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ⊆ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ )
62		ffvelcdm	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → ( 𝑓 ‘ 𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } )
63	61 62	sselid	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → ( 𝑓 ‘ 𝑚 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) )
64	63	adantrr	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( 𝑓 ‘ 𝑚 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) )
65		ffvelcdm	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → ( 𝑓 ‘ 𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } )
66	61 65	sselid	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → ( 𝑓 ‘ 𝑧 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) )
67	66	adantrl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( 𝑓 ‘ 𝑧 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) )
68	35	dyaddisj	⊢ ( ( ( 𝑓 ‘ 𝑚 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∧ ( 𝑓 ‘ 𝑧 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
69	64 67 68	syl2anc	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
70	52 69	sylan	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
71		df-3or	⊢ ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ↔ ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
72	70 71	sylib	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
73		elrabi	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( 𝑓 ‘ 𝑧 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } )
74		fveq2	⊢ ( 𝑎 = ( 𝑓 ‘ 𝑚 ) → ( [,] ‘ 𝑎 ) = ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) )
75	74	sseq1d	⊢ ( 𝑎 = ( 𝑓 ‘ 𝑚 ) → ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) ) )
76		eqeq1	⊢ ( 𝑎 = ( 𝑓 ‘ 𝑚 ) → ( 𝑎 = 𝑐 ↔ ( 𝑓 ‘ 𝑚 ) = 𝑐 ) )
77	75 76	imbi12d	⊢ ( 𝑎 = ( 𝑓 ‘ 𝑚 ) → ( ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑚 ) = 𝑐 ) ) )
78	77	ralbidv	⊢ ( 𝑎 = ( 𝑓 ‘ 𝑚 ) → ( ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑚 ) = 𝑐 ) ) )
79	78	elrab	⊢ ( ( 𝑓 ‘ 𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ↔ ( ( 𝑓 ‘ 𝑚 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∧ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑚 ) = 𝑐 ) ) )
80	79	simprbi	⊢ ( ( 𝑓 ‘ 𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑚 ) = 𝑐 ) )
81		fveq2	⊢ ( 𝑐 = ( 𝑓 ‘ 𝑧 ) → ( [,] ‘ 𝑐 ) = ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) )
82	81	sseq2d	⊢ ( 𝑐 = ( 𝑓 ‘ 𝑧 ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) )
83		eqeq2	⊢ ( 𝑐 = ( 𝑓 ‘ 𝑧 ) → ( ( 𝑓 ‘ 𝑚 ) = 𝑐 ↔ ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) )
84	82 83	imbi12d	⊢ ( 𝑐 = ( 𝑓 ‘ 𝑧 ) → ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑚 ) = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) ) )
85	84	rspcva	⊢ ( ( ( 𝑓 ‘ 𝑧 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∧ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑚 ) = 𝑐 ) ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) )
86	73 80 85	syl2anr	⊢ ( ( ( 𝑓 ‘ 𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑓 ‘ 𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) )
87		elrabi	⊢ ( ( 𝑓 ‘ 𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( 𝑓 ‘ 𝑚 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } )
88		fveq2	⊢ ( 𝑎 = ( 𝑓 ‘ 𝑧 ) → ( [,] ‘ 𝑎 ) = ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) )
89	88	sseq1d	⊢ ( 𝑎 = ( 𝑓 ‘ 𝑧 ) → ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) ) )
90		eqeq1	⊢ ( 𝑎 = ( 𝑓 ‘ 𝑧 ) → ( 𝑎 = 𝑐 ↔ ( 𝑓 ‘ 𝑧 ) = 𝑐 ) )
91	89 90	imbi12d	⊢ ( 𝑎 = ( 𝑓 ‘ 𝑧 ) → ( ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑧 ) = 𝑐 ) ) )
92	91	ralbidv	⊢ ( 𝑎 = ( 𝑓 ‘ 𝑧 ) → ( ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑧 ) = 𝑐 ) ) )
93	92	elrab	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ↔ ( ( 𝑓 ‘ 𝑧 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∧ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑧 ) = 𝑐 ) ) )
94	93	simprbi	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑧 ) = 𝑐 ) )
95		fveq2	⊢ ( 𝑐 = ( 𝑓 ‘ 𝑚 ) → ( [,] ‘ 𝑐 ) = ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) )
96	95	sseq2d	⊢ ( 𝑐 = ( 𝑓 ‘ 𝑚 ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) )
97		eqeq2	⊢ ( 𝑐 = ( 𝑓 ‘ 𝑚 ) → ( ( 𝑓 ‘ 𝑧 ) = 𝑐 ↔ ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) ) )
98	96 97	imbi12d	⊢ ( 𝑐 = ( 𝑓 ‘ 𝑚 ) → ( ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑧 ) = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) ) ) )
99	98	rspcva	⊢ ( ( ( 𝑓 ‘ 𝑚 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∧ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑧 ) = 𝑐 ) ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) ) )
100	87 94 99	syl2an	⊢ ( ( ( 𝑓 ‘ 𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑓 ‘ 𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) ) )
101		eqcom	⊢ ( ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) ↔ ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) )
102	100 101	imbitrdi	⊢ ( ( ( 𝑓 ‘ 𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑓 ‘ 𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) → ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) )
103	86 102	jaod	⊢ ( ( ( 𝑓 ‘ 𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑓 ‘ 𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) → ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) )
104	62 65 103	syl2an	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) ∧ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) ) → ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) → ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) )
105	104	anandis	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) → ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) )
106	52 105	sylan	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) → ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) )
107		f1of1	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → 𝑓 : ℕ –1-1→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } )
108		f1veqaeq	⊢ ( ( 𝑓 : ℕ –1-1→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) → 𝑚 = 𝑧 ) )
109	107 108	sylan	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) → 𝑚 = 𝑧 ) )
110	106 109	syld	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) → 𝑚 = 𝑧 ) )
111	110	orim1d	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∨ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) → ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ) )
112	72 111	mpd	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
113	112	ralrimivva	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
114		eqeq1	⊢ ( 𝑚 = 𝑧 → ( 𝑚 = 𝑝 ↔ 𝑧 = 𝑝 ) )
115		2fveq3	⊢ ( 𝑚 = 𝑧 → ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) = ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) )
116	115	ineq1d	⊢ ( 𝑚 = 𝑧 → ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) )
117	116	eqeq1d	⊢ ( 𝑚 = 𝑧 → ( ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ↔ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) )
118	114 117	orbi12d	⊢ ( 𝑚 = 𝑧 → ( ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ↔ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) )
119	118	ralbidv	⊢ ( 𝑚 = 𝑧 → ( ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ↔ ∀ 𝑝 ∈ ℕ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) )
120	119	cbvralvw	⊢ ( ∀ 𝑚 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ↔ ∀ 𝑧 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) )
121		eqeq2	⊢ ( 𝑧 = 𝑝 → ( 𝑚 = 𝑧 ↔ 𝑚 = 𝑝 ) )
122		2fveq3	⊢ ( 𝑧 = 𝑝 → ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) = ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) )
123	122	ineq2d	⊢ ( 𝑧 = 𝑝 → ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) )
124	123	eqeq1d	⊢ ( 𝑧 = 𝑝 → ( ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ↔ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) )
125	121 124	orbi12d	⊢ ( 𝑧 = 𝑝 → ( ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ↔ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) )
126	125	cbvralvw	⊢ ( ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ↔ ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) )
127	126	ralbii	⊢ ( ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ↔ ∀ 𝑚 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) )
128	122	disjor	⊢ ( Disj 𝑧 ∈ ℕ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ↔ ∀ 𝑧 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) )
129	120 127 128	3bitr4ri	⊢ ( Disj 𝑧 ∈ ℕ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ↔ ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
130	113 129	sylibr	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → Disj 𝑧 ∈ ℕ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) )
131		eqid	⊢ seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) )
132	60 130 131	uniiccvol	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( vol* ‘ ∪ ran ( [,] ∘ 𝑓 ) ) = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) )
133	132	adantl	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( vol* ‘ ∪ ran ( [,] ∘ 𝑓 ) ) = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) )
134	51 133	eqtr3d	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( vol* ‘ 𝐴 ) = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) )
135	18 134	breqtrd	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → 𝑀 < sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) )
136		absf	⊢ abs : ℂ ⟶ ℝ
137		subf	⊢ − : ( ℂ × ℂ ) ⟶ ℂ
138		fco	⊢ ( ( abs : ℂ ⟶ ℝ ∧ − : ( ℂ × ℂ ) ⟶ ℂ ) → ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ )
139	136 137 138	mp2an	⊢ ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ
140		zre	⊢ ( 𝑥 ∈ ℤ → 𝑥 ∈ ℝ )
141		2re	⊢ 2 ∈ ℝ
142		reexpcl	⊢ ( ( 2 ∈ ℝ ∧ 𝑦 ∈ ℕ0 ) → ( 2 ↑ 𝑦 ) ∈ ℝ )
143	141 142	mpan	⊢ ( 𝑦 ∈ ℕ0 → ( 2 ↑ 𝑦 ) ∈ ℝ )
144		2cn	⊢ 2 ∈ ℂ
145		2ne0	⊢ 2 ≠ 0
146		nn0z	⊢ ( 𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ )
147		expne0i	⊢ ( ( 2 ∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ ) → ( 2 ↑ 𝑦 ) ≠ 0 )
148	144 145 146 147	mp3an12i	⊢ ( 𝑦 ∈ ℕ0 → ( 2 ↑ 𝑦 ) ≠ 0 )
149	143 148	jca	⊢ ( 𝑦 ∈ ℕ0 → ( ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) )
150		redivcl	⊢ ( ( 𝑥 ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) → ( 𝑥 / ( 2 ↑ 𝑦 ) ) ∈ ℝ )
151		peano2re	⊢ ( 𝑥 ∈ ℝ → ( 𝑥 + 1 ) ∈ ℝ )
152		redivcl	⊢ ( ( ( 𝑥 + 1 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) → ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ∈ ℝ )
153	151 152	syl3an1	⊢ ( ( 𝑥 ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) → ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ∈ ℝ )
154	150 153	opelxpd	⊢ ( ( 𝑥 ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) → ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ∈ ( ℝ × ℝ ) )
155	154	3expb	⊢ ( ( 𝑥 ∈ ℝ ∧ ( ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) ) → ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ∈ ( ℝ × ℝ ) )
156	140 149 155	syl2an	⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ∈ ( ℝ × ℝ ) )
157	156	rgen2	⊢ ∀ 𝑥 ∈ ℤ ∀ 𝑦 ∈ ℕ0 ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ∈ ( ℝ × ℝ )
158		eqid	⊢ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) = ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ )
159	158	fmpo	⊢ ( ∀ 𝑥 ∈ ℤ ∀ 𝑦 ∈ ℕ0 ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ∈ ( ℝ × ℝ ) ↔ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) : ( ℤ × ℕ0 ) ⟶ ( ℝ × ℝ ) )
160	157 159	mpbi	⊢ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) : ( ℤ × ℕ0 ) ⟶ ( ℝ × ℝ )
161		frn	⊢ ( ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) : ( ℤ × ℕ0 ) ⟶ ( ℝ × ℝ ) → ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ⊆ ( ℝ × ℝ ) )
162	160 161	ax-mp	⊢ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ⊆ ( ℝ × ℝ )
163	42 162	sstri	⊢ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ⊆ ( ℝ × ℝ )
164	53 163	sstri	⊢ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ⊆ ( ℝ × ℝ )
165		ax-resscn	⊢ ℝ ⊆ ℂ
166		xpss12	⊢ ( ( ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ ) → ( ℝ × ℝ ) ⊆ ( ℂ × ℂ ) )
167	165 165 166	mp2an	⊢ ( ℝ × ℝ ) ⊆ ( ℂ × ℂ )
168	164 167	sstri	⊢ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ⊆ ( ℂ × ℂ )
169		fss	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ⊆ ( ℂ × ℂ ) ) → 𝑓 : ℕ ⟶ ( ℂ × ℂ ) )
170	168 169	mpan2	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → 𝑓 : ℕ ⟶ ( ℂ × ℂ ) )
171		fco	⊢ ( ( ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ ∧ 𝑓 : ℕ ⟶ ( ℂ × ℂ ) ) → ( ( abs ∘ − ) ∘ 𝑓 ) : ℕ ⟶ ℝ )
172	139 170 171	sylancr	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( ( abs ∘ − ) ∘ 𝑓 ) : ℕ ⟶ ℝ )
173		nnuz	⊢ ℕ = ( ℤ≥ ‘ 1 )
174		1z	⊢ 1 ∈ ℤ
175	174	a1i	⊢ ( ( ( abs ∘ − ) ∘ 𝑓 ) : ℕ ⟶ ℝ → 1 ∈ ℤ )
176		ffvelcdm	⊢ ( ( ( ( abs ∘ − ) ∘ 𝑓 ) : ℕ ⟶ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑛 ) ∈ ℝ )
177	173 175 176	serfre	⊢ ( ( ( abs ∘ − ) ∘ 𝑓 ) : ℕ ⟶ ℝ → seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) : ℕ ⟶ ℝ )
178		frn	⊢ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) : ℕ ⟶ ℝ → ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ⊆ ℝ )
179		ressxr	⊢ ℝ ⊆ ℝ*
180	178 179	sstrdi	⊢ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) : ℕ ⟶ ℝ → ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ⊆ ℝ* )
181	52 172 177 180	4syl	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ⊆ ℝ* )
182		rexr	⊢ ( 𝑀 ∈ ℝ → 𝑀 ∈ ℝ* )
183	182	3ad2ant2	⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) → 𝑀 ∈ ℝ* )
184		supxrlub	⊢ ( ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ⊆ ℝ* ∧ 𝑀 ∈ ℝ* ) → ( 𝑀 < sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ↔ ∃ 𝑧 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) 𝑀 < 𝑧 ) )
185	181 183 184	syl2anr	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( 𝑀 < sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ↔ ∃ 𝑧 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) 𝑀 < 𝑧 ) )
186	135 185	mpbid	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ∃ 𝑧 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) 𝑀 < 𝑧 )
187		seqfn	⊢ ( 1 ∈ ℤ → seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ( ℤ≥ ‘ 1 ) )
188	174 187	ax-mp	⊢ seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ( ℤ≥ ‘ 1 )
189	173	fneq2i	⊢ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ℕ ↔ seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ( ℤ≥ ‘ 1 ) )
190	188 189	mpbir	⊢ seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ℕ
191		breq2	⊢ ( 𝑧 = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) → ( 𝑀 < 𝑧 ↔ 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) )
192	191	rexrn	⊢ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ℕ → ( ∃ 𝑧 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) 𝑀 < 𝑧 ↔ ∃ 𝑛 ∈ ℕ 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) )
193	190 192	ax-mp	⊢ ( ∃ 𝑧 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) 𝑀 < 𝑧 ↔ ∃ 𝑛 ∈ ℕ 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
194	186 193	sylib	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ∃ 𝑛 ∈ ℕ 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
195	60	ffvelcdmda	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → ( 𝑓 ‘ 𝑧 ) ∈ ( ≤ ∩ ( ℝ × ℝ ) ) )
196		0le0	⊢ 0 ≤ 0
197		df-br	⊢ ( 0 ≤ 0 ↔ ⟨ 0 , 0 ⟩ ∈ ≤ )
198	196 197	mpbi	⊢ ⟨ 0 , 0 ⟩ ∈ ≤
199		0re	⊢ 0 ∈ ℝ
200		opelxpi	⊢ ( ( 0 ∈ ℝ ∧ 0 ∈ ℝ ) → ⟨ 0 , 0 ⟩ ∈ ( ℝ × ℝ ) )
201	199 199 200	mp2an	⊢ ⟨ 0 , 0 ⟩ ∈ ( ℝ × ℝ )
202		elin	⊢ ( ⟨ 0 , 0 ⟩ ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ↔ ( ⟨ 0 , 0 ⟩ ∈ ≤ ∧ ⟨ 0 , 0 ⟩ ∈ ( ℝ × ℝ ) ) )
203	198 201 202	mpbir2an	⊢ ⟨ 0 , 0 ⟩ ∈ ( ≤ ∩ ( ℝ × ℝ ) )
204		ifcl	⊢ ( ( ( 𝑓 ‘ 𝑧 ) ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ ⟨ 0 , 0 ⟩ ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ∈ ( ≤ ∩ ( ℝ × ℝ ) ) )
205	195 203 204	sylancl	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ∈ ( ≤ ∩ ( ℝ × ℝ ) ) )
206	205	fmpttd	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) )
207		df-ov	⊢ ( 0 (,) 0 ) = ( (,) ‘ ⟨ 0 , 0 ⟩ )
208		iooid	⊢ ( 0 (,) 0 ) = ∅
209	207 208	eqtr3i	⊢ ( (,) ‘ ⟨ 0 , 0 ⟩ ) = ∅
210	209	ineq1i	⊢ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ( ∅ ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) )
211		0in	⊢ ( ∅ ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅
212	210 211	eqtri	⊢ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅
213	212	olci	⊢ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ )
214		ineq1	⊢ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) )
215	214	eqeq1d	⊢ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ↔ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
216	215	orbi2d	⊢ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ↔ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ) )
217		ineq1	⊢ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) )
218	217	eqeq1d	⊢ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ↔ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
219	218	orbi2d	⊢ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ↔ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ) )
220	216 219	ifboth	⊢ ( ( ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ∧ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ) → ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
221	112 213 220	sylancl	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) )
222	209	ineq2i	⊢ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ∅ )
223		in0	⊢ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ∅ ) = ∅
224	222 223	eqtri	⊢ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ∅
225	224	olci	⊢ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ∅ )
226		ineq2	⊢ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) )
227	226	eqeq1d	⊢ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ↔ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
228	227	orbi2d	⊢ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ↔ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) ) )
229		ineq2	⊢ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) )
230	229	eqeq1d	⊢ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ∅ ↔ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
231	230	orbi2d	⊢ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ∅ ) ↔ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) ) )
232	228 231	ifboth	⊢ ( ( ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ∧ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ∅ ) ) → ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
233	221 225 232	sylancl	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ ) ) → ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
234	233	ralrimivva	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
235		disjeq2	⊢ ( ∀ 𝑚 ∈ ℕ ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( Disj 𝑚 ∈ ℕ ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) ↔ Disj 𝑚 ∈ ℕ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) )
236		eleq1w	⊢ ( 𝑧 = 𝑚 → ( 𝑧 ∈ ( 1 ... 𝑛 ) ↔ 𝑚 ∈ ( 1 ... 𝑛 ) ) )
237		fveq2	⊢ ( 𝑧 = 𝑚 → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) )
238	236 237	ifbieq1d	⊢ ( 𝑧 = 𝑚 → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , ⟨ 0 , 0 ⟩ ) )
239		eqid	⊢ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) )
240		fvex	⊢ ( 𝑓 ‘ 𝑚 ) ∈ V
241		opex	⊢ ⟨ 0 , 0 ⟩ ∈ V
242	240 241	ifex	⊢ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , ⟨ 0 , 0 ⟩ ) ∈ V
243	238 239 242	fvmpt	⊢ ( 𝑚 ∈ ℕ → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , ⟨ 0 , 0 ⟩ ) )
244	243	fveq2d	⊢ ( 𝑚 ∈ ℕ → ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = ( (,) ‘ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , ⟨ 0 , 0 ⟩ ) ) )
245		fvif	⊢ ( (,) ‘ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , ⟨ 0 , 0 ⟩ ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) )
246	244 245	eqtrdi	⊢ ( 𝑚 ∈ ℕ → ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) )
247	235 246	mprg	⊢ ( Disj 𝑚 ∈ ℕ ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) ↔ Disj 𝑚 ∈ ℕ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) )
248		eleq1w	⊢ ( 𝑚 = 𝑧 → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ 𝑧 ∈ ( 1 ... 𝑛 ) ) )
249	248 115	ifbieq1d	⊢ ( 𝑚 = 𝑧 → if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) )
250	249	disjor	⊢ ( Disj 𝑚 ∈ ℕ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ↔ ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
251	247 250	bitri	⊢ ( Disj 𝑚 ∈ ℕ ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) ↔ ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
252	234 251	sylibr	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → Disj 𝑚 ∈ ℕ ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) )
253		eqid	⊢ seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) = seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) )
254	206 252 253	uniiccvol	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( vol* ‘ ∪ ran ( [,] ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) , ℝ* , < ) )
255	254	adantr	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( vol* ‘ ∪ ran ( [,] ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) , ℝ* , < ) )
256		rexpssxrxp	⊢ ( ℝ × ℝ ) ⊆ ( ℝ* × ℝ* )
257	164 256	sstri	⊢ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ⊆ ( ℝ* × ℝ* )
258	257 65	sselid	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ* × ℝ* ) )
259		0xr	⊢ 0 ∈ ℝ*
260		opelxpi	⊢ ( ( 0 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ⟨ 0 , 0 ⟩ ∈ ( ℝ* × ℝ* ) )
261	259 259 260	mp2an	⊢ ⟨ 0 , 0 ⟩ ∈ ( ℝ* × ℝ* )
262		ifcl	⊢ ( ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ* × ℝ* ) ∧ ⟨ 0 , 0 ⟩ ∈ ( ℝ* × ℝ* ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ∈ ( ℝ* × ℝ* ) )
263	258 261 262	sylancl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ∈ ( ℝ* × ℝ* ) )
264		eqidd	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) )
265		iccf	⊢ [,] : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ*
266	265	a1i	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → [,] : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ* )
267	266	feqmptd	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → [,] = ( 𝑚 ∈ ( ℝ* × ℝ* ) ↦ ( [,] ‘ 𝑚 ) ) )
268		fveq2	⊢ ( 𝑚 = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) → ( [,] ‘ 𝑚 ) = ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) )
269	263 264 267 268	fmptco	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( [,] ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) )
270	52 269	syl	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( [,] ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) )
271	270	rneqd	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ran ( [,] ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) )
272	271	unieqd	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∪ ran ( [,] ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ∪ ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) )
273		peano2nn	⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ℕ )
274	273 173	eleqtrdi	⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ( ℤ≥ ‘ 1 ) )
275		fzouzsplit	⊢ ( ( 𝑛 + 1 ) ∈ ( ℤ≥ ‘ 1 ) → ( ℤ≥ ‘ 1 ) = ( ( 1 ..^ ( 𝑛 + 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) )
276	274 275	syl	⊢ ( 𝑛 ∈ ℕ → ( ℤ≥ ‘ 1 ) = ( ( 1 ..^ ( 𝑛 + 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) )
277	173 276	eqtrid	⊢ ( 𝑛 ∈ ℕ → ℕ = ( ( 1 ..^ ( 𝑛 + 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) )
278		nnz	⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℤ )
279		fzval3	⊢ ( 𝑛 ∈ ℤ → ( 1 ... 𝑛 ) = ( 1 ..^ ( 𝑛 + 1 ) ) )
280	278 279	syl	⊢ ( 𝑛 ∈ ℕ → ( 1 ... 𝑛 ) = ( 1 ..^ ( 𝑛 + 1 ) ) )
281	280	uneq1d	⊢ ( 𝑛 ∈ ℕ → ( ( 1 ... 𝑛 ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = ( ( 1 ..^ ( 𝑛 + 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) )
282	277 281	eqtr4d	⊢ ( 𝑛 ∈ ℕ → ℕ = ( ( 1 ... 𝑛 ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) )
283		fvif	⊢ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) )
284	283	a1i	⊢ ( 𝑛 ∈ ℕ → ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) )
285	282 284	iuneq12d	⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ℕ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ∪ 𝑧 ∈ ( ( 1 ... 𝑛 ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) )
286		fvex	⊢ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ∈ V
287	286	dfiun3	⊢ ∪ 𝑧 ∈ ℕ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ∪ ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) )
288		iunxun	⊢ ∪ 𝑧 ∈ ( ( 1 ... 𝑛 ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) )
289	285 287 288	3eqtr3g	⊢ ( 𝑛 ∈ ℕ → ∪ ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ) )
290		iftrue	⊢ ( 𝑧 ∈ ( 1 ... 𝑛 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) )
291	290	iuneq2i	⊢ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) )
292	291	a1i	⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) )
293		uznfz	⊢ ( 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) → ¬ 𝑧 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) )
294	293	adantl	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) → ¬ 𝑧 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) )
295		nncn	⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℂ )
296		ax-1cn	⊢ 1 ∈ ℂ
297		pncan	⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑛 + 1 ) − 1 ) = 𝑛 )
298	295 296 297	sylancl	⊢ ( 𝑛 ∈ ℕ → ( ( 𝑛 + 1 ) − 1 ) = 𝑛 )
299	298	oveq2d	⊢ ( 𝑛 ∈ ℕ → ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) = ( 1 ... 𝑛 ) )
300	299	eleq2d	⊢ ( 𝑛 ∈ ℕ → ( 𝑧 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) ↔ 𝑧 ∈ ( 1 ... 𝑛 ) ) )
301	300	notbid	⊢ ( 𝑛 ∈ ℕ → ( ¬ 𝑧 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) ↔ ¬ 𝑧 ∈ ( 1 ... 𝑛 ) ) )
302	301	adantr	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) → ( ¬ 𝑧 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) ↔ ¬ 𝑧 ∈ ( 1 ... 𝑛 ) ) )
303	294 302	mpbid	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) → ¬ 𝑧 ∈ ( 1 ... 𝑛 ) )
304	303	iffalsed	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ( [,] ‘ ⟨ 0 , 0 ⟩ ) )
305	304	iuneq2dv	⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) )
306	292 305	uneq12d	⊢ ( 𝑛 ∈ ℕ → ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ) = ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) )
307	289 306	eqtrd	⊢ ( 𝑛 ∈ ℕ → ∪ ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) )
308	272 307	sylan9eq	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ∪ ran ( [,] ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) )
309	308	fveq2d	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( vol* ‘ ∪ ran ( [,] ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) = ( vol* ‘ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ) )
310		xrltso	⊢ < Or ℝ*
311	310	a1i	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → < Or ℝ* )
312		elnnuz	⊢ ( 𝑛 ∈ ℕ ↔ 𝑛 ∈ ( ℤ≥ ‘ 1 ) )
313	312	biimpi	⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ( ℤ≥ ‘ 1 ) )
314	313	adantl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ( ℤ≥ ‘ 1 ) )
315		elfznn	⊢ ( 𝑢 ∈ ( 1 ... 𝑛 ) → 𝑢 ∈ ℕ )
316	172	ffvelcdmda	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑢 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑢 ) ∈ ℝ )
317	315 316	sylan2	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑢 ∈ ( 1 ... 𝑛 ) ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑢 ) ∈ ℝ )
318	317	adantlr	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑢 ∈ ( 1 ... 𝑛 ) ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑢 ) ∈ ℝ )
319		readdcl	⊢ ( ( 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ) → ( 𝑢 + 𝑣 ) ∈ ℝ )
320	319	adantl	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ ( 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ) ) → ( 𝑢 + 𝑣 ) ∈ ℝ )
321	314 318 320	seqcl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ )
322	321	rexrd	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ* )
323		eqidd	⊢ ( 𝑚 ∈ ( 1 ... 𝑛 ) → ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) )
324		iftrue	⊢ ( 𝑚 ∈ ( 1 ... 𝑛 ) → if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , ⟨ 0 , 0 ⟩ ) = ( 𝑓 ‘ 𝑚 ) )
325	238 324	sylan9eqr	⊢ ( ( 𝑚 ∈ ( 1 ... 𝑛 ) ∧ 𝑧 = 𝑚 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) = ( 𝑓 ‘ 𝑚 ) )
326		elfznn	⊢ ( 𝑚 ∈ ( 1 ... 𝑛 ) → 𝑚 ∈ ℕ )
327	240	a1i	⊢ ( 𝑚 ∈ ( 1 ... 𝑛 ) → ( 𝑓 ‘ 𝑚 ) ∈ V )
328	323 325 326 327	fvmptd	⊢ ( 𝑚 ∈ ( 1 ... 𝑛 ) → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = ( 𝑓 ‘ 𝑚 ) )
329	328	adantl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ( 1 ... 𝑛 ) ) → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = ( 𝑓 ‘ 𝑚 ) )
330	329	fveq2d	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ( 1 ... 𝑛 ) ) → ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑚 ) ) )
331		fvex	⊢ ( 𝑓 ‘ 𝑧 ) ∈ V
332	331 241	ifex	⊢ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ∈ V
333	332 239	fnmpti	⊢ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) Fn ℕ
334		fvco2	⊢ ( ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) Fn ℕ ∧ 𝑚 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) )
335	333 326 334	sylancr	⊢ ( 𝑚 ∈ ( 1 ... 𝑛 ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) )
336	335	adantl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ( 1 ... 𝑛 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) )
337		ffn	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → 𝑓 Fn ℕ )
338		fvco2	⊢ ( ( 𝑓 Fn ℕ ∧ 𝑚 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑚 ) ) )
339	337 326 338	syl2an	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ( 1 ... 𝑛 ) ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑚 ) ) )
340	330 336 339	3eqtr4d	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ( 1 ... 𝑛 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) )
341	340	adantlr	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑚 ∈ ( 1 ... 𝑛 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) )
342	314 341	seqfveq	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
343	174	a1i	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → 1 ∈ ℤ )
344	168 65	sselid	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → ( 𝑓 ‘ 𝑧 ) ∈ ( ℂ × ℂ ) )
345		0cn	⊢ 0 ∈ ℂ
346		opelxpi	⊢ ( ( 0 ∈ ℂ ∧ 0 ∈ ℂ ) → ⟨ 0 , 0 ⟩ ∈ ( ℂ × ℂ ) )
347	345 345 346	mp2an	⊢ ⟨ 0 , 0 ⟩ ∈ ( ℂ × ℂ )
348		ifcl	⊢ ( ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℂ × ℂ ) ∧ ⟨ 0 , 0 ⟩ ∈ ( ℂ × ℂ ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ∈ ( ℂ × ℂ ) )
349	344 347 348	sylancl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ∈ ( ℂ × ℂ ) )
350	349	fmpttd	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) : ℕ ⟶ ( ℂ × ℂ ) )
351		fco	⊢ ( ( ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ ∧ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) : ℕ ⟶ ( ℂ × ℂ ) ) → ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) : ℕ ⟶ ℝ )
352	139 350 351	sylancr	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) : ℕ ⟶ ℝ )
353	352	ffvelcdmda	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) ∈ ℝ )
354	173 343 353	serfre	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) : ℕ ⟶ ℝ )
355	354	ffnd	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) Fn ℕ )
356		fnfvelrn	⊢ ( ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) Fn ℕ ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) )
357	355 356	sylan	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) )
358	342 357	eqeltrrd	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) )
359	354	frnd	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ⊆ ℝ )
360	359	adantr	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ⊆ ℝ )
361	360	sselda	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑚 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ) → 𝑚 ∈ ℝ )
362	321	adantr	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑚 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ )
363		readdcl	⊢ ( ( 𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ) → ( 𝑚 + 𝑢 ) ∈ ℝ )
364	363	adantl	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ ( 𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ) ) → ( 𝑚 + 𝑢 ) ∈ ℝ )
365		recn	⊢ ( 𝑚 ∈ ℝ → 𝑚 ∈ ℂ )
366		recn	⊢ ( 𝑢 ∈ ℝ → 𝑢 ∈ ℂ )
367		recn	⊢ ( 𝑣 ∈ ℝ → 𝑣 ∈ ℂ )
368		addass	⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ ) → ( ( 𝑚 + 𝑢 ) + 𝑣 ) = ( 𝑚 + ( 𝑢 + 𝑣 ) ) )
369	365 366 367 368	syl3an	⊢ ( ( 𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ) → ( ( 𝑚 + 𝑢 ) + 𝑣 ) = ( 𝑚 + ( 𝑢 + 𝑣 ) ) )
370	369	adantl	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ ( 𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ) ) → ( ( 𝑚 + 𝑢 ) + 𝑣 ) = ( 𝑚 + ( 𝑢 + 𝑣 ) ) )
371		nnltp1le	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) → ( 𝑛 < 𝑡 ↔ ( 𝑛 + 1 ) ≤ 𝑡 ) )
372	371	biimpa	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( 𝑛 + 1 ) ≤ 𝑡 )
373	273	nnzd	⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ℤ )
374		nnz	⊢ ( 𝑡 ∈ ℕ → 𝑡 ∈ ℤ )
375		eluz	⊢ ( ( ( 𝑛 + 1 ) ∈ ℤ ∧ 𝑡 ∈ ℤ ) → ( 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ↔ ( 𝑛 + 1 ) ≤ 𝑡 ) )
376	373 374 375	syl2an	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) → ( 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ↔ ( 𝑛 + 1 ) ≤ 𝑡 ) )
377	376	adantr	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ↔ ( 𝑛 + 1 ) ≤ 𝑡 ) )
378	372 377	mpbird	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) )
379	378	adantlll	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) )
380	313	ad3antlr	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → 𝑛 ∈ ( ℤ≥ ‘ 1 ) )
381		simplll	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } )
382		elfznn	⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚 ∈ ℕ )
383	381 382 353	syl2an	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) ∈ ℝ )
384	364 370 379 380 383	seqsplit	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) = ( ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) + ( seq ( 𝑛 + 1 ) ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ) )
385	342	ad2antrr	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
386		elfzelz	⊢ ( 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) → 𝑚 ∈ ℤ )
387	386	adantl	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℤ )
388		0red	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 ∈ ℝ )
389	273	nnred	⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ℝ )
390	389	ad3antrrr	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑛 + 1 ) ∈ ℝ )
391	386	zred	⊢ ( 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) → 𝑚 ∈ ℝ )
392	391	adantl	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℝ )
393	273	nngt0d	⊢ ( 𝑛 ∈ ℕ → 0 < ( 𝑛 + 1 ) )
394	393	ad3antrrr	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 < ( 𝑛 + 1 ) )
395		elfzle1	⊢ ( 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) → ( 𝑛 + 1 ) ≤ 𝑚 )
396	395	adantl	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑛 + 1 ) ≤ 𝑚 )
397	388 390 392 394 396	ltletrd	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 < 𝑚 )
398		elnnz	⊢ ( 𝑚 ∈ ℕ ↔ ( 𝑚 ∈ ℤ ∧ 0 < 𝑚 ) )
399	387 397 398	sylanbrc	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℕ )
400	333 399 334	sylancr	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) )
401		eqidd	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) )
402		nnre	⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ )
403	402	adantr	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑛 ∈ ℝ )
404	389	adantr	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑛 + 1 ) ∈ ℝ )
405	391	adantl	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℝ )
406	402	ltp1d	⊢ ( 𝑛 ∈ ℕ → 𝑛 < ( 𝑛 + 1 ) )
407	406	adantr	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑛 < ( 𝑛 + 1 ) )
408	395	adantl	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑛 + 1 ) ≤ 𝑚 )
409	403 404 405 407 408	ltletrd	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑛 < 𝑚 )
410	409	adantr	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → 𝑛 < 𝑚 )
411	403 405	ltnled	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛 ) )
412		breq1	⊢ ( 𝑚 = 𝑧 → ( 𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛 ) )
413	412	equcoms	⊢ ( 𝑧 = 𝑚 → ( 𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛 ) )
414	413	notbid	⊢ ( 𝑧 = 𝑚 → ( ¬ 𝑚 ≤ 𝑛 ↔ ¬ 𝑧 ≤ 𝑛 ) )
415	411 414	sylan9bb	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → ( 𝑛 < 𝑚 ↔ ¬ 𝑧 ≤ 𝑛 ) )
416	410 415	mpbid	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → ¬ 𝑧 ≤ 𝑛 )
417		elfzle2	⊢ ( 𝑧 ∈ ( 1 ... 𝑛 ) → 𝑧 ≤ 𝑛 )
418	416 417	nsyl	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → ¬ 𝑧 ∈ ( 1 ... 𝑛 ) )
419	418	iffalsed	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) = ⟨ 0 , 0 ⟩ )
420	386	adantl	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℤ )
421		0red	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 ∈ ℝ )
422	393	adantr	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 < ( 𝑛 + 1 ) )
423	421 404 405 422 408	ltletrd	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 < 𝑚 )
424	420 423 398	sylanbrc	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℕ )
425	241	a1i	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ⟨ 0 , 0 ⟩ ∈ V )
426	401 419 424 425	fvmptd	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = ⟨ 0 , 0 ⟩ )
427	426	ad4ant14	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = ⟨ 0 , 0 ⟩ )
428	427	fveq2d	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = ( ( abs ∘ − ) ‘ ⟨ 0 , 0 ⟩ ) )
429	400 428	eqtrd	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ⟨ 0 , 0 ⟩ ) )
430		fvco3	⊢ ( ( − : ( ℂ × ℂ ) ⟶ ℂ ∧ ⟨ 0 , 0 ⟩ ∈ ( ℂ × ℂ ) ) → ( ( abs ∘ − ) ‘ ⟨ 0 , 0 ⟩ ) = ( abs ‘ ( − ‘ ⟨ 0 , 0 ⟩ ) ) )
431	137 347 430	mp2an	⊢ ( ( abs ∘ − ) ‘ ⟨ 0 , 0 ⟩ ) = ( abs ‘ ( − ‘ ⟨ 0 , 0 ⟩ ) )
432		df-ov	⊢ ( 0 − 0 ) = ( − ‘ ⟨ 0 , 0 ⟩ )
433		0m0e0	⊢ ( 0 − 0 ) = 0
434	432 433	eqtr3i	⊢ ( − ‘ ⟨ 0 , 0 ⟩ ) = 0
435	434	fveq2i	⊢ ( abs ‘ ( − ‘ ⟨ 0 , 0 ⟩ ) ) = ( abs ‘ 0 )
436		abs0	⊢ ( abs ‘ 0 ) = 0
437	435 436	eqtri	⊢ ( abs ‘ ( − ‘ ⟨ 0 , 0 ⟩ ) ) = 0
438	431 437	eqtri	⊢ ( ( abs ∘ − ) ‘ ⟨ 0 , 0 ⟩ ) = 0
439	429 438	eqtrdi	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = 0 )
440		elfzuz	⊢ ( 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) → 𝑚 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) )
441		c0ex	⊢ 0 ∈ V
442	441	fvconst2	⊢ ( 𝑚 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) → ( ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ‘ 𝑚 ) = 0 )
443	440 442	syl	⊢ ( 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) → ( ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ‘ 𝑚 ) = 0 )
444	443	adantl	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ‘ 𝑚 ) = 0 )
445	439 444	eqtr4d	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ‘ 𝑚 ) )
446	378 445	seqfveq	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq ( 𝑛 + 1 ) ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) = ( seq ( 𝑛 + 1 ) ( + , ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ) ‘ 𝑡 ) )
447		eqid	⊢ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) = ( ℤ≥ ‘ ( 𝑛 + 1 ) )
448	447	ser0	⊢ ( 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) → ( seq ( 𝑛 + 1 ) ( + , ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ) ‘ 𝑡 ) = 0 )
449	378 448	syl	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq ( 𝑛 + 1 ) ( + , ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ) ‘ 𝑡 ) = 0 )
450	446 449	eqtrd	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq ( 𝑛 + 1 ) ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) = 0 )
451	450	adantlll	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq ( 𝑛 + 1 ) ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) = 0 )
452	385 451	oveq12d	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) + ( seq ( 𝑛 + 1 ) ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ) = ( ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) + 0 ) )
453	172	ffvelcdmda	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) ∈ ℝ )
454	326 453	sylan2	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ( 1 ... 𝑛 ) ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) ∈ ℝ )
455	454	adantlr	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑚 ∈ ( 1 ... 𝑛 ) ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) ∈ ℝ )
456		readdcl	⊢ ( ( 𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ ) → ( 𝑚 + 𝑣 ) ∈ ℝ )
457	456	adantl	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ ( 𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ ) ) → ( 𝑚 + 𝑣 ) ∈ ℝ )
458	314 455 457	seqcl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ )
459	458	ad2antrr	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ )
460	459	recnd	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℂ )
461	460	addridd	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) + 0 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
462	452 461	eqtrd	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) + ( seq ( 𝑛 + 1 ) ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
463	384 462	eqtrd	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
464	453	ad5ant15	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) ∈ ℝ )
465	326 464	sylan2	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( 1 ... 𝑛 ) ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) ∈ ℝ )
466	380 465 364	seqcl	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ )
467	466	leidd	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
468	463 467	eqbrtrd	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
469		elnnuz	⊢ ( 𝑡 ∈ ℕ ↔ 𝑡 ∈ ( ℤ≥ ‘ 1 ) )
470	469	biimpi	⊢ ( 𝑡 ∈ ℕ → 𝑡 ∈ ( ℤ≥ ‘ 1 ) )
471	470	ad2antlr	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) → 𝑡 ∈ ( ℤ≥ ‘ 1 ) )
472		eqidd	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) )
473		simpr	⊢ ( ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → 𝑧 = 𝑚 )
474		elfzle1	⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → 1 ≤ 𝑚 )
475	474	adantl	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 1 ≤ 𝑚 )
476	382	nnred	⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚 ∈ ℝ )
477	476	adantl	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ∈ ℝ )
478		nnre	⊢ ( 𝑡 ∈ ℕ → 𝑡 ∈ ℝ )
479	478	ad3antlr	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑡 ∈ ℝ )
480	402	ad3antrrr	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑛 ∈ ℝ )
481		elfzle2	⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚 ≤ 𝑡 )
482	481	adantl	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ≤ 𝑡 )
483		simplr	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑡 ≤ 𝑛 )
484	477 479 480 482 483	letrd	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ≤ 𝑛 )
485		elfzelz	⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚 ∈ ℤ )
486	278	ad2antrr	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) → 𝑛 ∈ ℤ )
487		elfz	⊢ ( ( 𝑚 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ ( 1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛 ) ) )
488	174 487	mp3an2	⊢ ( ( 𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ ( 1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛 ) ) )
489	485 486 488	syl2anr	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ ( 1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛 ) ) )
490	475 484 489	mpbir2and	⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ∈ ( 1 ... 𝑛 ) )
491	490	ad5ant2345	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ∈ ( 1 ... 𝑛 ) )
492	491	adantr	⊢ ( ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → 𝑚 ∈ ( 1 ... 𝑛 ) )
493	473 492	eqeltrd	⊢ ( ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → 𝑧 ∈ ( 1 ... 𝑛 ) )
494		iftrue	⊢ ( 𝑧 ∈ ( 1 ... 𝑛 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) = ( 𝑓 ‘ 𝑧 ) )
495	493 494	syl	⊢ ( ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) = ( 𝑓 ‘ 𝑧 ) )
496	237	adantl	⊢ ( ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) )
497	495 496	eqtrd	⊢ ( ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) = ( 𝑓 ‘ 𝑚 ) )
498	382	adantl	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ∈ ℕ )
499	240	a1i	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( 𝑓 ‘ 𝑚 ) ∈ V )
500	472 497 498 499	fvmptd	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = ( 𝑓 ‘ 𝑚 ) )
501	500	fveq2d	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑚 ) ) )
502	333 382 334	sylancr	⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) )
503	502	adantl	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) )
504		simplll	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) → 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } )
505		fvco3	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑚 ) ) )
506	504 382 505	syl2an	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑚 ) ) )
507	501 503 506	3eqtr4d	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) )
508	471 507	seqfveq	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑡 ) )
509		eluz	⊢ ( ( 𝑡 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑡 ) ↔ 𝑡 ≤ 𝑛 ) )
510	374 278 509	syl2anr	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑡 ) ↔ 𝑡 ≤ 𝑛 ) )
511	510	biimpar	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑡 ) )
512	511	adantlll	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑡 ) )
513	504 326 453	syl2an	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑛 ) ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) ∈ ℝ )
514		elfzelz	⊢ ( 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) → 𝑚 ∈ ℤ )
515	514	adantl	⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 𝑚 ∈ ℤ )
516		0red	⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 0 ∈ ℝ )
517		peano2nn	⊢ ( 𝑡 ∈ ℕ → ( 𝑡 + 1 ) ∈ ℕ )
518	517	nnred	⊢ ( 𝑡 ∈ ℕ → ( 𝑡 + 1 ) ∈ ℝ )
519	518	adantr	⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → ( 𝑡 + 1 ) ∈ ℝ )
520	514	zred	⊢ ( 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) → 𝑚 ∈ ℝ )
521	520	adantl	⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 𝑚 ∈ ℝ )
522	517	nngt0d	⊢ ( 𝑡 ∈ ℕ → 0 < ( 𝑡 + 1 ) )
523	522	adantr	⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 0 < ( 𝑡 + 1 ) )
524		elfzle1	⊢ ( 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) → ( 𝑡 + 1 ) ≤ 𝑚 )
525	524	adantl	⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → ( 𝑡 + 1 ) ≤ 𝑚 )
526	516 519 521 523 525	ltletrd	⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 0 < 𝑚 )
527	515 526 398	sylanbrc	⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 𝑚 ∈ ℕ )
528	527	adantlr	⊢ ( ( ( 𝑡 ∈ ℕ ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 𝑚 ∈ ℕ )
529	528	adantlll	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 𝑚 ∈ ℕ )
530	170	ffvelcdmda	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → ( 𝑓 ‘ 𝑚 ) ∈ ( ℂ × ℂ ) )
531		ffvelcdm	⊢ ( ( − : ( ℂ × ℂ ) ⟶ ℂ ∧ ( 𝑓 ‘ 𝑚 ) ∈ ( ℂ × ℂ ) ) → ( − ‘ ( 𝑓 ‘ 𝑚 ) ) ∈ ℂ )
532	137 530 531	sylancr	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → ( − ‘ ( 𝑓 ‘ 𝑚 ) ) ∈ ℂ )
533	532	absge0d	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → 0 ≤ ( abs ‘ ( − ‘ ( 𝑓 ‘ 𝑚 ) ) ) )
534		fvco3	⊢ ( ( − : ( ℂ × ℂ ) ⟶ ℂ ∧ ( 𝑓 ‘ 𝑚 ) ∈ ( ℂ × ℂ ) ) → ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑚 ) ) = ( abs ‘ ( − ‘ ( 𝑓 ‘ 𝑚 ) ) ) )
535	137 530 534	sylancr	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑚 ) ) = ( abs ‘ ( − ‘ ( 𝑓 ‘ 𝑚 ) ) ) )
536	505 535	eqtrd	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) = ( abs ‘ ( − ‘ ( 𝑓 ‘ 𝑚 ) ) ) )
537	533 536	breqtrrd	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑚 ∈ ℕ ) → 0 ≤ ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) )
538	537	ad5ant15	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ℕ ) → 0 ≤ ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) )
539	529 538	syldan	⊢ ( ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 0 ≤ ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) )
540	471 512 513 539	sermono	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
541	508 540	eqbrtrd	⊢ ( ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
542	402	ad2antlr	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) → 𝑛 ∈ ℝ )
543	478	adantl	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) → 𝑡 ∈ ℝ )
544	468 541 542 543	ltlecasei	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑡 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
545	544	ralrimiva	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ∀ 𝑡 ∈ ℕ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
546		breq1	⊢ ( 𝑚 = ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) → ( 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) )
547	546	ralrn	⊢ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) Fn ℕ → ( ∀ 𝑚 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ∀ 𝑡 ∈ ℕ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) )
548	355 547	syl	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( ∀ 𝑚 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ∀ 𝑡 ∈ ℕ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) )
549	548	adantr	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( ∀ 𝑚 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ∀ 𝑡 ∈ ℕ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) )
550	545 549	mpbird	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ∀ 𝑚 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
551	550	r19.21bi	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑚 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ) → 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
552	361 362 551	lensymd	⊢ ( ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) ∧ 𝑚 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ) → ¬ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) < 𝑚 )
553	311 322 358 552	supmax	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) , ℝ* , < ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
554	52 553	sylan	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) , ℝ* , < ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
555	255 309 554	3eqtr3rd	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) = ( vol* ‘ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ) )
556		elfznn	⊢ ( 𝑧 ∈ ( 1 ... 𝑛 ) → 𝑧 ∈ ℕ )
557	164 65	sselid	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) )
558		1st2nd2	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( 𝑓 ‘ 𝑧 ) = ⟨ ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ⟩ )
559	558	fveq2d	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) = ( [,] ‘ ⟨ ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ⟩ ) )
560		df-ov	⊢ ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ( [,] ‘ ⟨ ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ⟩ )
561	559 560	eqtr4di	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) = ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) )
562		xp1st	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ )
563		xp2nd	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ )
564		iccssre	⊢ ( ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) ⊆ ℝ )
565	562 563 564	syl2anc	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) ⊆ ℝ )
566	561 565	eqsstrd	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ )
567	557 566	syl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ )
568	52 556 567	syl2an	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ( 1 ... 𝑛 ) ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ )
569	568	ralrimiva	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ )
570		iunss	⊢ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ ↔ ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ )
571	569 570	sylibr	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ )
572	571	adantr	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ )
573		uzid	⊢ ( ( 𝑛 + 1 ) ∈ ℤ → ( 𝑛 + 1 ) ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) )
574		ne0i	⊢ ( ( 𝑛 + 1 ) ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) → ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ≠ ∅ )
575		iunconst	⊢ ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ≠ ∅ → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) = ( [,] ‘ ⟨ 0 , 0 ⟩ ) )
576	373 573 574 575	4syl	⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) = ( [,] ‘ ⟨ 0 , 0 ⟩ ) )
577		iccid	⊢ ( 0 ∈ ℝ* → ( 0 [,] 0 ) = { 0 } )
578	259 577	ax-mp	⊢ ( 0 [,] 0 ) = { 0 }
579		df-ov	⊢ ( 0 [,] 0 ) = ( [,] ‘ ⟨ 0 , 0 ⟩ )
580	578 579	eqtr3i	⊢ { 0 } = ( [,] ‘ ⟨ 0 , 0 ⟩ )
581	576 580	eqtr4di	⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) = { 0 } )
582		snssi	⊢ ( 0 ∈ ℝ → { 0 } ⊆ ℝ )
583	199 582	ax-mp	⊢ { 0 } ⊆ ℝ
584	581 583	eqsstrdi	⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ⊆ ℝ )
585	584	adantl	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ⊆ ℝ )
586	581	fveq2d	⊢ ( 𝑛 ∈ ℕ → ( vol* ‘ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ( vol* ‘ { 0 } ) )
587	586	adantl	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( vol* ‘ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ( vol* ‘ { 0 } ) )
588		ovolsn	⊢ ( 0 ∈ ℝ → ( vol* ‘ { 0 } ) = 0 )
589	199 588	ax-mp	⊢ ( vol* ‘ { 0 } ) = 0
590	587 589	eqtrdi	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( vol* ‘ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = 0 )
591		ovolunnul	⊢ ( ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ ∧ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ⊆ ℝ ∧ ( vol* ‘ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = 0 ) → ( vol* ‘ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ) = ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) )
592	572 585 590 591	syl3anc	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( vol* ‘ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ) = ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) )
593	555 592	eqtrd	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) = ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) )
594	593	breq2d	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) )
595	594	biimpd	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑛 ∈ ℕ ) → ( 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) → 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) )
596	595	reximdva	⊢ ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( ∃ 𝑛 ∈ ℕ 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) → ∃ 𝑛 ∈ ℕ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) )
597	596	adantl	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( ∃ 𝑛 ∈ ℕ 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) → ∃ 𝑛 ∈ ℕ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) )
598	194 597	mpd	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ∃ 𝑛 ∈ ℕ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) )
599		fzfi	⊢ ( 1 ... 𝑛 ) ∈ Fin
600		icccld	⊢ ( ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
601	562 563 600	syl2anc	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
602	561 601	eqeltrd	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
603	557 602	syl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
604	556 603	sylan2	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ( 1 ... 𝑛 ) ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
605	604	ralrimiva	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
606		uniretop	⊢ ℝ = ∪ ( topGen ‘ ran (,) )
607	606	iuncld	⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( 1 ... 𝑛 ) ∈ Fin ∧ ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ) → ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
608	1 599 605 607	mp3an12i	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
609	608	adantr	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑛 ∈ ℕ ∧ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) → ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
610		fveq2	⊢ ( 𝑏 = ( 𝑓 ‘ 𝑧 ) → ( [,] ‘ 𝑏 ) = ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) )
611	610	sseq1d	⊢ ( 𝑏 = ( 𝑓 ‘ 𝑧 ) → ( ( [,] ‘ 𝑏 ) ⊆ 𝐴 ↔ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 ) )
612	611	elrab	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ↔ ( ( 𝑓 ‘ 𝑧 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∧ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 ) )
613	612	simprbi	⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 )
614	65 73 613	3syl	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ℕ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 )
615	556 614	sylan2	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ 𝑧 ∈ ( 1 ... 𝑛 ) ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 )
616	615	ralrimiva	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 )
617		iunss	⊢ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 ↔ ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 )
618	616 617	sylibr	⊢ ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 )
619	618	adantr	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑛 ∈ ℕ ∧ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) → ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 )
620		simprr	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑛 ∈ ℕ ∧ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) → 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) )
621		sseq1	⊢ ( 𝑠 = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( 𝑠 ⊆ 𝐴 ↔ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 ) )
622		fveq2	⊢ ( 𝑠 = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( vol* ‘ 𝑠 ) = ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) )
623	622	breq2d	⊢ ( 𝑠 = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( 𝑀 < ( vol* ‘ 𝑠 ) ↔ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) )
624	621 623	anbi12d	⊢ ( 𝑠 = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) ↔ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) )
625	624	rspcev	⊢ ( ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ∧ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )
626	609 619 620 625	syl12anc	⊢ ( ( 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑛 ∈ ℕ ∧ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )
627	52 626	sylan	⊢ ( ( 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑛 ∈ ℕ ∧ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )
628	627	adantll	⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) ∧ ( 𝑛 ∈ ℕ ∧ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )
629	598 628	rexlimddv	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )
630	629	adantlr	⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑓 : ℕ –1-1-onto→ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )
631	17 630	exlimddv	⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 ≠ ∅ ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )
632	15 631	pm2.61dane	⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) )