Theorem fzval 9031

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 NN_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 e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( 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 3788	. . . 4 |- ( m = M -> ( m <_ k <-> M <_ k ) )
2	1	anbi1d 452	. . 3 |- ( m = M -> ( ( m <_ k /\ k <_ n ) <-> ( M <_ k /\ k <_ n ) ) )
3	2	rabbidv 2593	. 2 |- ( m = M -> { k e. ZZ | ( m <_ k /\ k <_ n ) } = { k e. ZZ | ( M <_ k /\ k <_ n ) } )
4		breq2 3789	. . . 4 |- ( n = N -> ( k <_ n <-> k <_ N ) )
5	4	anbi2d 451	. . 3 |- ( n = N -> ( ( M <_ k /\ k <_ n ) <-> ( M <_ k /\ k <_ N ) ) )
6	5	rabbidv 2593	. 2 |- ( n = N -> { k e. ZZ | ( M <_ k /\ k <_ n ) } = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
7		df-fz 9030	. 2 |- ... = ( m e. ZZ , n e. ZZ |-> { k e. ZZ | ( m <_ k /\ k <_ n ) } )
8		zex 8360	. . 3 |- ZZ e. _V
9	8	rabex 3922	. 2 |- { k e. ZZ | ( M <_ k /\ k <_ N ) } e. _V
10	3, 6, 7, 9	ovmpt2 5656	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   {crab 2352   class class class wbr 3785  (class class class)co 5532   <_ cle 7154   ZZcz 8351   ...cfz 9029

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280  ax-cnex 7067  ax-resscn 7068

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-neg 7282  df-z 8352  df-fz 9030

This theorem is referenced by:  fzval2  9032  elfz1  9034  fznlem  9060