Theorem fzval 13518

Description: The value of a finite set of sequential integers. E.g., 2...5 means the set {2, 3, 4, 5}. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where ℕ_k means our 1...𝑘; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
fzval	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})

Distinct variable groups:   𝑘,𝑀   𝑘,𝑁

Proof of Theorem fzval

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5151	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑚 ≤ 𝑘 ↔ 𝑀 ≤ 𝑘))
2	1	anbi1d 630	. . 3 ⊢ (𝑚 = 𝑀 → ((𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)))
3	2	rabbidv 3437	. 2 ⊢ (𝑚 = 𝑀 → {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
4		breq2 5152	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑘 ≤ 𝑛 ↔ 𝑘 ≤ 𝑁))
5	4	anbi2d 629	. . 3 ⊢ (𝑛 = 𝑁 → ((𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)))
6	5	rabbidv 3437	. 2 ⊢ (𝑛 = 𝑁 → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
7		df-fz 13517	. 2 ⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
8		zex 12597	. . 3 ⊢ ℤ ∈ V
9	8	rabex 5334	. 2 ⊢ {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∈ V
10	3, 6, 7, 9	ovmpo 7581	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3429   class class class wbr 5148  (class class class)co 7420   ≤ cle 11279  ℤcz 12588  ...cfz 13516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-cnex 11194  ax-resscn 11195

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-neg 11477  df-z 12589  df-fz 13517

This theorem is referenced by:  fzval2  13519  elfz1  13521