Theorem fzval 10012

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 4008	. . . 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 2728	. 2 |- ( m = M -> { k e. ZZ | ( m <_ k /\ k <_ n ) } = { k e. ZZ | ( M <_ k /\ k <_ n ) } )
4		breq2 4009	. . . 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 2728	. 2 |- ( n = N -> { k e. ZZ | ( M <_ k /\ k <_ n ) } = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
7		df-fz 10011	. 2 |- ... = ( m e. ZZ , n e. ZZ |-> { k e. ZZ | ( m <_ k /\ k <_ n ) } )
8		zex 9264	. . 3 |- ZZ e. _V
9	8	rabex 4149	. 2 |- { k e. ZZ | ( M <_ k /\ k <_ N ) } e. _V
10	3, 6, 7, 9	ovmpo 6012	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   {crab 2459   class class class wbr 4005  (class class class)co 5877   <_ cle 7995   ZZcz 9255   ...cfz 10010

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-cnex 7904  ax-resscn 7905

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-neg 8133  df-z 9256  df-fz 10011

This theorem is referenced by:  fzval2  10013  elfz1  10015  fznlem  10043