Theorem iccpval 41351

Description: Partition consisting of a fixed number M of parts. (Contributed by AV, 9-Jul-2020.)

Assertion

Ref	Expression
iccpval	|- ( M e. NN -> (RePart ` M ) = { p e. ( RR* ^m ( 0 ... M ) ) | A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) } )

Distinct variable group:   i, p, M

Proof of Theorem iccpval

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iccp 41350	. . 3 |- RePart = ( m e. NN |-> { p e. ( RR* ^m ( 0 ... m ) ) | A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) } )
2	1	a1i 11	. 2 |- ( M e. NN -> RePart = ( m e. NN |-> { p e. ( RR* ^m ( 0 ... m ) ) | A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) } ) )
3		oveq2 6658	. . . . 5 |- ( m = M -> ( 0 ... m ) = ( 0 ... M ) )
4	3	oveq2d 6666	. . . 4 |- ( m = M -> ( RR* ^m ( 0 ... m ) ) = ( RR* ^m ( 0 ... M ) ) )
5		oveq2 6658	. . . . 5 |- ( m = M -> ( 0..^ m ) = ( 0..^ M ) )
6	5	raleqdv 3144	. . . 4 |- ( m = M -> ( A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) <-> A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) )
7	4, 6	rabeqbidv 3195	. . 3 |- ( m = M -> { p e. ( RR* ^m ( 0 ... m ) ) | A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) } = { p e. ( RR* ^m ( 0 ... M ) ) | A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) } )
8	7	adantl 482	. 2 |- ( ( M e. NN /\ m = M ) -> { p e. ( RR* ^m ( 0 ... m ) ) | A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) } = { p e. ( RR* ^m ( 0 ... M ) ) | A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) } )
9		id 22	. 2 |- ( M e. NN -> M e. NN )
10		ovex 6678	. . . 4 |- ( RR* ^m ( 0 ... M ) ) e. _V
11	10	rabex 4813	. . 3 |- { p e. ( RR* ^m ( 0 ... M ) ) | A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) } e. _V
12	11	a1i 11	. 2 |- ( M e. NN -> { p e. ( RR* ^m ( 0 ... M ) ) | A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) } e. _V )
13	2, 8, 9, 12	fvmptd 6288	1 |- ( M e. NN -> (RePart ` M ) = { p e. ( RR* ^m ( 0 ... M ) ) | A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   NNcn 11020   ...cfz 12326  ..^cfzo 12465  RePartciccp 41349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-iccp 41350

This theorem is referenced by:  iccpart  41352