Theorem suppssr 6932

Description: A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)

Hypotheses

Ref	Expression
suppssr.f	|- ( ph -> F : A --> B )
suppssr.n	|- ( ph -> ( F supp Z ) C_ W )
suppssr.a	|- ( ph -> A e. V )
suppssr.z	|- ( ph -> Z e. U )

Assertion

Ref	Expression
suppssr	|- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )

Proof of Theorem suppssr

Step	Hyp	Ref	Expression
1		eldif 3486	. 2 |- ( X e. ( A \ W ) <-> ( X e. A /\ -. X e. W ) )
2		fvex 5876	. . . . . 6 |- ( F ` X ) e. _V
3		eldifsn 4152	. . . . . 6 |- ( ( F ` X ) e. ( _V \ { Z } ) <-> ( ( F ` X ) e. _V /\ ( F ` X ) =/= Z ) )
4	2, 3	mpbiran 916	. . . . 5 |- ( ( F ` X ) e. ( _V \ { Z } ) <-> ( F ` X ) =/= Z )
5		suppssr.f	. . . . . . . . . 10 |- ( ph -> F : A --> B )
6		ffn 5731	. . . . . . . . . 10 |- ( F : A --> B -> F Fn A )
7	5, 6	syl 16	. . . . . . . . 9 |- ( ph -> F Fn A )
8		suppssr.a	. . . . . . . . 9 |- ( ph -> A e. V )
9		suppssr.z	. . . . . . . . 9 |- ( ph -> Z e. U )
10		elsuppfn 6910	. . . . . . . . 9 |- ( ( F Fn A /\ A e. V /\ Z e. U ) -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) =/= Z ) ) )
11	7, 8, 9, 10	syl3anc 1228	. . . . . . . 8 |- ( ph -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) =/= Z ) ) )
12		ibar 504	. . . . . . . . . . 11 |- ( ( F ` X ) e. _V -> ( ( F ` X ) =/= Z <-> ( ( F ` X ) e. _V /\ ( F ` X ) =/= Z ) ) )
13	2, 12	mp1i 12	. . . . . . . . . 10 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z <-> ( ( F ` X ) e. _V /\ ( F ` X ) =/= Z ) ) )
14	13, 3	syl6bbr 263	. . . . . . . . 9 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z <-> ( F ` X ) e. ( _V \ { Z } ) ) )
15	14	pm5.32da 641	. . . . . . . 8 |- ( ph -> ( ( X e. A /\ ( F ` X ) =/= Z ) <-> ( X e. A /\ ( F ` X ) e. ( _V \ { Z } ) ) ) )
16	11, 15	bitrd 253	. . . . . . 7 |- ( ph -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) e. ( _V \ { Z } ) ) ) )
17		suppssr.n	. . . . . . . 8 |- ( ph -> ( F supp Z ) C_ W )
18	17	sseld 3503	. . . . . . 7 |- ( ph -> ( X e. ( F supp Z ) -> X e. W ) )
19	16, 18	sylbird 235	. . . . . 6 |- ( ph -> ( ( X e. A /\ ( F ` X ) e. ( _V \ { Z } ) ) -> X e. W ) )
20	19	expdimp 437	. . . . 5 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) e. ( _V \ { Z } ) -> X e. W ) )
21	4, 20	syl5bir 218	. . . 4 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z -> X e. W ) )
22	21	necon1bd 2685	. . 3 |- ( ( ph /\ X e. A ) -> ( -. X e. W -> ( F ` X ) = Z ) )
23	22	impr 619	. 2 |- ( ( ph /\ ( X e. A /\ -. X e. W ) ) -> ( F ` X ) = Z )
24	1, 23	sylan2b 475	1 |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   _Vcvv 3113   \ cdif 3473   C_ wss 3476   {csn 4027   Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285   supp csupp 6902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-supp 6903

This theorem is referenced by:  fsuppmptif  7860  fsuppco2  7863  fsuppcor  7864  cantnfp1lem1  8098  cantnfp1lem3  8100  cantnflem1d  8108  cantnflem1  8109  cnfcom2lem  8146  gsumval3  16726  gsumcllem  16727  gsumzaddlem  16749  gsumzmhm  16772  gsumpt  16803  gsum2dlem1  16812  gsum2dlem2  16813  gsum2d  16814  dprdfinv  16873  dprdfadd  16874  dmdprdsplitlem  16898  dpjidcl  16921  gsumdixp  17071  lcomfsupp  17362  psrbaglesupp  17828  psrbagaddcl  17831  psrbaglefi  17834  mplsubglem  17904  mpllsslem  17905  mplsubrglem  17911  mplmonmul  17937  mplcoe1  17938  mplcoe5  17942  mplbas2  17945  evlslem4  17985  evlslem2  17991  uvcresum  18631  frlmsslsp  18636  rrxcph  21651  rrxmval  21659  rrxmetlem  21661  rrxmet  21662  rrxdstprj1  21663  deg1mul3le  22344  eulerpartlemb  28058