Theorem sniffsupp 8315

Description: A function mapping all but one arguments to zero is finitely supported. (Contributed by AV, 8-Jul-2019.)

Hypotheses

Ref	Expression
sniffsupp.i	|- ( ph -> I e. V )
sniffsupp.0	|- ( ph -> .0. e. W )
sniffsupp.f	|- F = ( x e. I |-> if ( x = X , A , .0. ) )

Assertion

Ref	Expression
sniffsupp	|- ( ph -> F finSupp .0. )

Distinct variable groups:   x, I   x, X   x, .0.   ph, x

Allowed substitution hints:   A( x)   F( x)   V( x)   W( x)

Proof of Theorem sniffsupp

Step	Hyp	Ref	Expression
1		sniffsupp.f	. 2 |- F = ( x e. I |-> if ( x = X , A , .0. ) )
2		snfi 8038	. . . 4 |- { X } e. Fin
3		eldifsni 4320	. . . . . . . 8 |- ( x e. ( I \ { X } ) -> x =/= X )
4	3	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. ( I \ { X } ) ) -> x =/= X )
5	4	neneqd 2799	. . . . . 6 |- ( ( ph /\ x e. ( I \ { X } ) ) -> -. x = X )
6	5	iffalsed 4097	. . . . 5 |- ( ( ph /\ x e. ( I \ { X } ) ) -> if ( x = X , A , .0. ) = .0. )
7		sniffsupp.i	. . . . 5 |- ( ph -> I e. V )
8	6, 7	suppss2 7329	. . . 4 |- ( ph -> ( ( x e. I |-> if ( x = X , A , .0. ) ) supp .0. ) C_ { X } )
9		ssfi 8180	. . . 4 |- ( ( { X } e. Fin /\ ( ( x e. I |-> if ( x = X , A , .0. ) ) supp .0. ) C_ { X } ) -> ( ( x e. I |-> if ( x = X , A , .0. ) ) supp .0. ) e. Fin )
10	2, 8, 9	sylancr 695	. . 3 |- ( ph -> ( ( x e. I |-> if ( x = X , A , .0. ) ) supp .0. ) e. Fin )
11		funmpt 5926	. . . . 5 |- Fun ( x e. I |-> if ( x = X , A , .0. ) )
12	11	a1i 11	. . . 4 |- ( ph -> Fun ( x e. I |-> if ( x = X , A , .0. ) ) )
13		mptexg 6484	. . . . 5 |- ( I e. V -> ( x e. I |-> if ( x = X , A , .0. ) ) e. _V )
14	7, 13	syl 17	. . . 4 |- ( ph -> ( x e. I |-> if ( x = X , A , .0. ) ) e. _V )
15		sniffsupp.0	. . . 4 |- ( ph -> .0. e. W )
16		funisfsupp 8280	. . . 4 |- ( ( Fun ( x e. I |-> if ( x = X , A , .0. ) ) /\ ( x e. I |-> if ( x = X , A , .0. ) ) e. _V /\ .0. e. W ) -> ( ( x e. I |-> if ( x = X , A , .0. ) ) finSupp .0. <-> ( ( x e. I |-> if ( x = X , A , .0. ) ) supp .0. ) e. Fin ) )
17	12, 14, 15, 16	syl3anc 1326	. . 3 |- ( ph -> ( ( x e. I |-> if ( x = X , A , .0. ) ) finSupp .0. <-> ( ( x e. I |-> if ( x = X , A , .0. ) ) supp .0. ) e. Fin ) )
18	10, 17	mpbird 247	. 2 |- ( ph -> ( x e. I |-> if ( x = X , A , .0. ) ) finSupp .0. )
19	1, 18	syl5eqbr 4688	1 |- ( ph -> F finSupp .0. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   Fun wfun 5882  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275

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-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-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-supp 7296  df-1o 7560  df-er 7742  df-en 7956  df-fin 7959  df-fsupp 8276

This theorem is referenced by:  dprdfid  18416  snifpsrbag  19366