Theorem reprinrn 30696

Description: Representations with term in an intersection. (Contributed by Thierry Arnoux, 11-Dec-2021.)

Hypotheses

Ref	Expression
reprval.a	|- ( ph -> A C_ NN )
reprval.m	|- ( ph -> M e. ZZ )
reprval.s	|- ( ph -> S e. NN0 )

Assertion

Ref	Expression
reprinrn	|- ( ph -> ( c e. ( ( A i^i B ) (repr ` S ) M ) <-> ( c e. ( A (repr ` S ) M ) /\ ran c C_ B ) ) )

Distinct variable groups:   A, c   M, c   S, c   ph, c   B, c

Proof of Theorem reprinrn

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin 6085	. . . . 5 |- ( c : ( 0..^ S ) --> ( A i^i B ) <-> ( c : ( 0..^ S ) --> A /\ c : ( 0..^ S ) --> B ) )
2		df-f 5892	. . . . . . 7 |- ( c : ( 0..^ S ) --> B <-> ( c Fn ( 0..^ S ) /\ ran c C_ B ) )
3		ffn 6045	. . . . . . . . . 10 |- ( c : ( 0..^ S ) --> A -> c Fn ( 0..^ S ) )
4	3	adantl 482	. . . . . . . . 9 |- ( ( ph /\ c : ( 0..^ S ) --> A ) -> c Fn ( 0..^ S ) )
5	4	biantrurd 529	. . . . . . . 8 |- ( ( ph /\ c : ( 0..^ S ) --> A ) -> ( ran c C_ B <-> ( c Fn ( 0..^ S ) /\ ran c C_ B ) ) )
6	5	bicomd 213	. . . . . . 7 |- ( ( ph /\ c : ( 0..^ S ) --> A ) -> ( ( c Fn ( 0..^ S ) /\ ran c C_ B ) <-> ran c C_ B ) )
7	2, 6	syl5bb 272	. . . . . 6 |- ( ( ph /\ c : ( 0..^ S ) --> A ) -> ( c : ( 0..^ S ) --> B <-> ran c C_ B ) )
8	7	pm5.32da 673	. . . . 5 |- ( ph -> ( ( c : ( 0..^ S ) --> A /\ c : ( 0..^ S ) --> B ) <-> ( c : ( 0..^ S ) --> A /\ ran c C_ B ) ) )
9	1, 8	syl5bb 272	. . . 4 |- ( ph -> ( c : ( 0..^ S ) --> ( A i^i B ) <-> ( c : ( 0..^ S ) --> A /\ ran c C_ B ) ) )
10		nnex 11026	. . . . . . . 8 |- NN e. _V
11	10	a1i 11	. . . . . . 7 |- ( ph -> NN e. _V )
12		reprval.a	. . . . . . 7 |- ( ph -> A C_ NN )
13	11, 12	ssexd 4805	. . . . . 6 |- ( ph -> A e. _V )
14		inex1g 4801	. . . . . 6 |- ( A e. _V -> ( A i^i B ) e. _V )
15	13, 14	syl 17	. . . . 5 |- ( ph -> ( A i^i B ) e. _V )
16		ovex 6678	. . . . 5 |- ( 0..^ S ) e. _V
17		elmapg 7870	. . . . 5 |- ( ( ( A i^i B ) e. _V /\ ( 0..^ S ) e. _V ) -> ( c e. ( ( A i^i B ) ^m ( 0..^ S ) ) <-> c : ( 0..^ S ) --> ( A i^i B ) ) )
18	15, 16, 17	sylancl 694	. . . 4 |- ( ph -> ( c e. ( ( A i^i B ) ^m ( 0..^ S ) ) <-> c : ( 0..^ S ) --> ( A i^i B ) ) )
19		elmapg 7870	. . . . . 6 |- ( ( A e. _V /\ ( 0..^ S ) e. _V ) -> ( c e. ( A ^m ( 0..^ S ) ) <-> c : ( 0..^ S ) --> A ) )
20	13, 16, 19	sylancl 694	. . . . 5 |- ( ph -> ( c e. ( A ^m ( 0..^ S ) ) <-> c : ( 0..^ S ) --> A ) )
21	20	anbi1d 741	. . . 4 |- ( ph -> ( ( c e. ( A ^m ( 0..^ S ) ) /\ ran c C_ B ) <-> ( c : ( 0..^ S ) --> A /\ ran c C_ B ) ) )
22	9, 18, 21	3bitr4d 300	. . 3 |- ( ph -> ( c e. ( ( A i^i B ) ^m ( 0..^ S ) ) <-> ( c e. ( A ^m ( 0..^ S ) ) /\ ran c C_ B ) ) )
23	22	anbi1d 741	. 2 |- ( ph -> ( ( c e. ( ( A i^i B ) ^m ( 0..^ S ) ) /\ sum_ a e. ( 0..^ S ) ( c ` a ) = M ) <-> ( ( c e. ( A ^m ( 0..^ S ) ) /\ ran c C_ B ) /\ sum_ a e. ( 0..^ S ) ( c ` a ) = M ) ) )
24		inss1 3833	. . . . . 6 |- ( A i^i B ) C_ A
25	24, 12	syl5ss 3614	. . . . 5 |- ( ph -> ( A i^i B ) C_ NN )
26		reprval.m	. . . . 5 |- ( ph -> M e. ZZ )
27		reprval.s	. . . . 5 |- ( ph -> S e. NN0 )
28	25, 26, 27	reprval 30688	. . . 4 |- ( ph -> ( ( A i^i B ) (repr ` S ) M ) = { c e. ( ( A i^i B ) ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } )
29	28	eleq2d 2687	. . 3 |- ( ph -> ( c e. ( ( A i^i B ) (repr ` S ) M ) <-> c e. { c e. ( ( A i^i B ) ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } ) )
30		rabid 3116	. . 3 |- ( c e. { c e. ( ( A i^i B ) ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } <-> ( c e. ( ( A i^i B ) ^m ( 0..^ S ) ) /\ sum_ a e. ( 0..^ S ) ( c ` a ) = M ) )
31	29, 30	syl6bb 276	. 2 |- ( ph -> ( c e. ( ( A i^i B ) (repr ` S ) M ) <-> ( c e. ( ( A i^i B ) ^m ( 0..^ S ) ) /\ sum_ a e. ( 0..^ S ) ( c ` a ) = M ) ) )
32	12, 26, 27	reprval 30688	. . . . . 6 |- ( ph -> ( A (repr ` S ) M ) = { c e. ( A ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } )
33	32	eleq2d 2687	. . . . 5 |- ( ph -> ( c e. ( A (repr ` S ) M ) <-> c e. { c e. ( A ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } ) )
34		rabid 3116	. . . . 5 |- ( c e. { c e. ( A ^m ( 0..^ S ) ) | sum_ a e. ( 0..^ S ) ( c ` a ) = M } <-> ( c e. ( A ^m ( 0..^ S ) ) /\ sum_ a e. ( 0..^ S ) ( c ` a ) = M ) )
35	33, 34	syl6bb 276	. . . 4 |- ( ph -> ( c e. ( A (repr ` S ) M ) <-> ( c e. ( A ^m ( 0..^ S ) ) /\ sum_ a e. ( 0..^ S ) ( c ` a ) = M ) ) )
36	35	anbi1d 741	. . 3 |- ( ph -> ( ( c e. ( A (repr ` S ) M ) /\ ran c C_ B ) <-> ( ( c e. ( A ^m ( 0..^ S ) ) /\ sum_ a e. ( 0..^ S ) ( c ` a ) = M ) /\ ran c C_ B ) ) )
37		an32 839	. . 3 |- ( ( ( c e. ( A ^m ( 0..^ S ) ) /\ sum_ a e. ( 0..^ S ) ( c ` a ) = M ) /\ ran c C_ B ) <-> ( ( c e. ( A ^m ( 0..^ S ) ) /\ ran c C_ B ) /\ sum_ a e. ( 0..^ S ) ( c ` a ) = M ) )
38	36, 37	syl6bb 276	. 2 |- ( ph -> ( ( c e. ( A (repr ` S ) M ) /\ ran c C_ B ) <-> ( ( c e. ( A ^m ( 0..^ S ) ) /\ ran c C_ B ) /\ sum_ a e. ( 0..^ S ) ( c ` a ) = M ) ) )
39	23, 31, 38	3bitr4d 300	1 |- ( ph -> ( c e. ( ( A i^i B ) (repr ` S ) M ) <-> ( c e. ( A (repr ` S ) M ) /\ ran c C_ B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   NNcn 11020   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   sum_csu 14416  reprcrepr 30686

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-map 7859  df-neg 10269  df-nn 11021  df-z 11378  df-seq 12802  df-sum 14417  df-repr 30687

This theorem is referenced by:  hashreprin  30698