Theorem dprdff 18411

Description: A finitely supported function in S is a function into the base. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)

Hypotheses

Ref	Expression
dprdff.w	|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
dprdff.1	|- ( ph -> G dom DProd S )
dprdff.2	|- ( ph -> dom S = I )
dprdff.3	|- ( ph -> F e. W )
dprdff.b	|- B = ( Base ` G )

Assertion

Ref	Expression
dprdff	|- ( ph -> F : I --> B )

Distinct variable groups:   h, F   h, i, I   .0. , h   S, h, i

Allowed substitution hints:   ph( h, i)   B( h, i)   F( i)   G( h, i)   W( h, i)   .0. ( i)

Proof of Theorem dprdff

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dprdff.3	. . . 4 |- ( ph -> F e. W )
2		dprdff.w	. . . . 5 |- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
3		dprdff.1	. . . . 5 |- ( ph -> G dom DProd S )
4		dprdff.2	. . . . 5 |- ( ph -> dom S = I )
5	2, 3, 4	dprdw 18409	. . . 4 |- ( ph -> ( F e. W <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) ) )
6	1, 5	mpbid 222	. . 3 |- ( ph -> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) )
7	6	simp1d 1073	. 2 |- ( ph -> F Fn I )
8	6	simp2d 1074	. . 3 |- ( ph -> A. x e. I ( F ` x ) e. ( S ` x ) )
9	3, 4	dprdf2 18406	. . . . . . 7 |- ( ph -> S : I --> (SubGrp ` G ) )
10	9	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( S ` x ) e. (SubGrp ` G ) )
11		dprdff.b	. . . . . . 7 |- B = ( Base ` G )
12	11	subgss 17595	. . . . . 6 |- ( ( S ` x ) e. (SubGrp ` G ) -> ( S ` x ) C_ B )
13	10, 12	syl 17	. . . . 5 |- ( ( ph /\ x e. I ) -> ( S ` x ) C_ B )
14	13	sseld 3602	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( F ` x ) e. ( S ` x ) -> ( F ` x ) e. B ) )
15	14	ralimdva 2962	. . 3 |- ( ph -> ( A. x e. I ( F ` x ) e. ( S ` x ) -> A. x e. I ( F ` x ) e. B ) )
16	8, 15	mpd 15	. 2 |- ( ph -> A. x e. I ( F ` x ) e. B )
17		ffnfv 6388	. 2 |- ( F : I --> B <-> ( F Fn I /\ A. x e. I ( F ` x ) e. B ) )
18	7, 16, 17	sylanbrc 698	1 |- ( ph -> F : I --> B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   class class class wbr 4653   dom cdm 5114   Fn wfn 5883   -->wf 5884   ` cfv 5888   X_cixp 7908   finSupp cfsupp 8275   Basecbs 15857  SubGrpcsubg 17588   DProd cdprd 18392

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

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-ne 2795  df-nel 2898  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  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-1st 7168  df-2nd 7169  df-ixp 7909  df-subg 17591  df-dprd 18394

This theorem is referenced by:  dprdfcntz  18414  dprdssv  18415  dprdfid  18416  dprdfinv  18418  dprdfadd  18419  dprdfsub  18420  dprdfeq0  18421  dprdf11  18422  dprdlub  18425  dmdprdsplitlem  18436  dprddisj2  18438  dpjidcl  18457