Theorem fzval 10134

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 4048	. . . 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 2761	. 2 |- ( m = M -> { k e. ZZ | ( m <_ k /\ k <_ n ) } = { k e. ZZ | ( M <_ k /\ k <_ n ) } )
4		breq2 4049	. . . 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 2761	. 2 |- ( n = N -> { k e. ZZ | ( M <_ k /\ k <_ n ) } = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
7		df-fz 10133	. 2 |- ... = ( m e. ZZ , n e. ZZ |-> { k e. ZZ | ( m <_ k /\ k <_ n ) } )
8		zex 9383	. . 3 |- ZZ e. _V
9	8	rabex 4189	. 2 |- { k e. ZZ | ( M <_ k /\ k <_ N ) } e. _V
10	3, 6, 7, 9	ovmpo 6083	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   {crab 2488   class class class wbr 4045  (class class class)co 5946   <_ cle 8110   ZZcz 9374   ...cfz 10132

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-setind 4586  ax-cnex 8018  ax-resscn 8019

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-neg 8248  df-z 9375  df-fz 10133

This theorem is referenced by:  fzval2  10135  elfz1  10137  fznlem  10165