Theorem fzval 13482

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 5150	. . . 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 3440	. 2 ⊢ (𝑚 = 𝑀 → {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
4		breq2 5151	. . . 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 3440	. 2 ⊢ (𝑛 = 𝑁 → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
7		df-fz 13481	. 2 ⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
8		zex 12563	. . 3 ⊢ ℤ ∈ V
9	8	rabex 5331	. 2 ⊢ {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∈ V
10	3, 6, 7, 9	ovmpo 7564	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   class class class wbr 5147  (class class class)co 7405   ≤ cle 11245  ℤcz 12554  ...cfz 13480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-cnex 11162  ax-resscn 11163

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-neg 11443  df-z 12555  df-fz 13481

This theorem is referenced by:  fzval2  13483  elfz1  13485