Theorem dprdw 18409

Description: The property of being a finitely supported function in the family S. (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 )

Assertion

Ref	Expression
dprdw	|- ( ph -> ( F e. W <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) ) )

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

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

Proof of Theorem dprdw

Step	Hyp	Ref	Expression
1		elex 3212	. . . . 5 |- ( F e. X_ i e. I ( S ` i ) -> F e. _V )
2	1	a1i 11	. . . 4 |- ( ph -> ( F e. X_ i e. I ( S ` i ) -> F e. _V ) )
3		dprdff.1	. . . . . . 7 |- ( ph -> G dom DProd S )
4		dprdff.2	. . . . . . 7 |- ( ph -> dom S = I )
5	3, 4	dprddomcld 18400	. . . . . 6 |- ( ph -> I e. _V )
6		fnex 6481	. . . . . . 7 |- ( ( F Fn I /\ I e. _V ) -> F e. _V )
7	6	expcom 451	. . . . . 6 |- ( I e. _V -> ( F Fn I -> F e. _V ) )
8	5, 7	syl 17	. . . . 5 |- ( ph -> ( F Fn I -> F e. _V ) )
9	8	adantrd 484	. . . 4 |- ( ph -> ( ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) -> F e. _V ) )
10		fveq2 6191	. . . . . . . . 9 |- ( i = x -> ( S ` i ) = ( S ` x ) )
11	10	cbvixpv 7926	. . . . . . . 8 |- X_ i e. I ( S ` i ) = X_ x e. I ( S ` x )
12	11	eleq2i 2693	. . . . . . 7 |- ( F e. X_ i e. I ( S ` i ) <-> F e. X_ x e. I ( S ` x ) )
13		elixp2 7912	. . . . . . 7 |- ( F e. X_ x e. I ( S ` x ) <-> ( F e. _V /\ F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) )
14		3anass 1042	. . . . . . 7 |- ( ( F e. _V /\ F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) <-> ( F e. _V /\ ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) )
15	12, 13, 14	3bitri 286	. . . . . 6 |- ( F e. X_ i e. I ( S ` i ) <-> ( F e. _V /\ ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) )
16	15	baib 944	. . . . 5 |- ( F e. _V -> ( F e. X_ i e. I ( S ` i ) <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) )
17	16	a1i 11	. . . 4 |- ( ph -> ( F e. _V -> ( F e. X_ i e. I ( S ` i ) <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) ) )
18	2, 9, 17	pm5.21ndd 369	. . 3 |- ( ph -> ( F e. X_ i e. I ( S ` i ) <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) )
19	18	anbi1d 741	. 2 |- ( ph -> ( ( F e. X_ i e. I ( S ` i ) /\ F finSupp .0. ) <-> ( ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) /\ F finSupp .0. ) ) )
20		breq1 4656	. . 3 |- ( h = F -> ( h finSupp .0. <-> F finSupp .0. ) )
21		dprdff.w	. . 3 |- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
22	20, 21	elrab2 3366	. 2 |- ( F e. W <-> ( F e. X_ i e. I ( S ` i ) /\ F finSupp .0. ) )
23		df-3an 1039	. 2 |- ( ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) <-> ( ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) /\ F finSupp .0. ) )
24	19, 22, 23	3bitr4g 303	1 |- ( ph -> ( F e. W <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   class class class wbr 4653   dom cdm 5114   Fn wfn 5883   ` cfv 5888   X_cixp 7908   finSupp cfsupp 8275   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-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-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-oprab 6654  df-mpt2 6655  df-ixp 7909  df-dprd 18394

This theorem is referenced by:  dprdff  18411  dprdfcl  18412  dprdffsupp  18413  dprdsubg  18423