Theorem fzval 9823

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 3940	. . . 4 |- ( m = M -> ( m <_ k <-> M <_ k ) )
2	1	anbi1d 461	. . 3 |- ( m = M -> ( ( m <_ k /\ k <_ n ) <-> ( M <_ k /\ k <_ n ) ) )
3	2	rabbidv 2678	. 2 |- ( m = M -> { k e. ZZ | ( m <_ k /\ k <_ n ) } = { k e. ZZ | ( M <_ k /\ k <_ n ) } )
4		breq2 3941	. . . 4 |- ( n = N -> ( k <_ n <-> k <_ N ) )
5	4	anbi2d 460	. . 3 |- ( n = N -> ( ( M <_ k /\ k <_ n ) <-> ( M <_ k /\ k <_ N ) ) )
6	5	rabbidv 2678	. 2 |- ( n = N -> { k e. ZZ | ( M <_ k /\ k <_ n ) } = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
7		df-fz 9822	. 2 |- ... = ( m e. ZZ , n e. ZZ |-> { k e. ZZ | ( m <_ k /\ k <_ n ) } )
8		zex 9087	. . 3 |- ZZ e. _V
9	8	rabex 4080	. 2 |- { k e. ZZ | ( M <_ k /\ k <_ N ) } e. _V
10	3, 6, 7, 9	ovmpo 5914	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   {crab 2421   class class class wbr 3937  (class class class)co 5782   <_ cle 7825   ZZcz 9078   ...cfz 9821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-setind 4460  ax-cnex 7735  ax-resscn 7736

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-neg 7960  df-z 9079  df-fz 9822

This theorem is referenced by:  fzval2  9824  elfz1  9826  fznlem  9852