Theorem fzval 11703

Description: The value of a finite set of sequential integers. E.g., 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 _k means our ; 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

Distinct variable groups:   ,M   ,N

Proof of Theorem fzval

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4455	. . . 4
2	1	anbi1d 704	. . 3
3	2	rabbidv 3101	. 2
4		breq2 4456	. . . 4
5	4	anbi2d 703	. . 3
6	5	rabbidv 3101	. 2
7		df-fz 11702	. 2
8		zex 10898	. . 3
9	8	rabex 4603	. 2
10	3, 6, 7, 9	ovmpt2 6438	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {crab 2811   class class class wbr 4452  (class class class)co 6296  cle 9650  cz 10889  cfz 11701

This theorem is referenced by:  fzval2  11704  elfz1  11706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-cnex 9569  ax-resscn 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-neg 9831  df-z 10890  df-fz 11702