Theorem fzval 10392

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 4128	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑚 ≤ 𝑘 ↔ 𝑀 ≤ 𝑘))
2	1	anbi1d 469	. . 3 ⊢ (𝑚 = 𝑀 → ((𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)))
3	2	rabbidv 2810	. 2 ⊢ (𝑚 = 𝑀 → {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
4		breq2 4129	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑘 ≤ 𝑛 ↔ 𝑘 ≤ 𝑁))
5	4	anbi2d 468	. . 3 ⊢ (𝑛 = 𝑁 → ((𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)))
6	5	rabbidv 2810	. 2 ⊢ (𝑛 = 𝑁 → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
7		df-fz 10391	. 2 ⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
8		zex 9632	. . 3 ⊢ ℤ ∈ V
9	8	rabex 4275	. 2 ⊢ {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∈ V
10	3, 6, 7, 9	ovmpo 6214	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {crab 2532   class class class wbr 4125  (class class class)co 6075   ≤ cle 8351  ℤcz 9623  ...cfz 10390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679  ax-cnex 8260  ax-resscn 8261

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-neg 8490  df-z 9624  df-fz 10391

This theorem is referenced by:  fzval2  10393  elfz1  10395  fznlem  10424