Theorem fzval 10206

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 4086	. . . 4 |- ( m = M -> ( m <_ k <-> M <_ k ) )
2	1	anbi1d 465	. . 3 |- ( m = M -> ( ( m <_ k /\ k <_ n ) <-> ( M <_ k /\ k <_ n ) ) )
3	2	rabbidv 2788	. 2 |- ( m = M -> { k e. ZZ | ( m <_ k /\ k <_ n ) } = { k e. ZZ | ( M <_ k /\ k <_ n ) } )
4		breq2 4087	. . . 4 |- ( n = N -> ( k <_ n <-> k <_ N ) )
5	4	anbi2d 464	. . 3 |- ( n = N -> ( ( M <_ k /\ k <_ n ) <-> ( M <_ k /\ k <_ N ) ) )
6	5	rabbidv 2788	. 2 |- ( n = N -> { k e. ZZ | ( M <_ k /\ k <_ n ) } = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
7		df-fz 10205	. 2 |- ... = ( m e. ZZ , n e. ZZ |-> { k e. ZZ | ( m <_ k /\ k <_ n ) } )
8		zex 9455	. . 3 |- ZZ e. _V
9	8	rabex 4228	. 2 |- { k e. ZZ | ( M <_ k /\ k <_ N ) } e. _V
10	3, 6, 7, 9	ovmpo 6140	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   {crab 2512   class class class wbr 4083  (class class class)co 6001   <_ cle 8182   ZZcz 9446   ...cfz 10204

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629  ax-cnex 8090  ax-resscn 8091

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-neg 8320  df-z 9447  df-fz 10205

This theorem is referenced by:  fzval2  10207  elfz1  10209  fznlem  10237