Theorem suppssr 6720

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 3338	. 2 |- ( X e. ( A \ W ) <-> ( X e. A /\ -. X e. W ) )
2		fvex 5701	. . . . . 6 |- ( F ` X ) e. _V
3		eldifsn 4000	. . . . . 6 |- ( ( F ` X ) e. ( _V \ { Z } ) <-> ( ( F ` X ) e. _V /\ ( F ` X ) =/= Z ) )
4	2, 3	mpbiran 909	. . . . 5 |- ( ( F ` X ) e. ( _V \ { Z } ) <-> ( F ` X ) =/= Z )
5		suppssr.f	. . . . . . . . . 10 |- ( ph -> F : A --> B )
6		ffn 5559	. . . . . . . . . 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 6698	. . . . . . . . 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 1218	. . . . . . . 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 3355	. . . . . . 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 2679	. . 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 1369   e. wcel 1756   =/= wne 2606   _Vcvv 2972   \ cdif 3325   C_ wss 3328   {csn 3877   Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   supp csupp 6690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-supp 6691

This theorem is referenced by:  fsuppmptif  7649  fsuppco2  7652  fsuppcor  7653  cantnfp1lem1  7886  cantnfp1lem3  7888  cantnflem1d  7896  cantnflem1  7897  cnfcom2lem  7934  gsumval3  16385  gsumcllem  16386  gsumzaddlem  16408  gsumzmhm  16430  gsumpt  16454  gsum2dlem1  16461  gsum2dlem2  16462  gsum2d  16463  dprdfinv  16509  dprdfadd  16510  dmdprdsplitlem  16534  dpjidcl  16557  gsumdixp  16701  lcomfsupp  16985  psrbaglesupp  17435  psrbagaddcl  17438  psrbaglefi  17441  mplsubglem  17510  mpllsslem  17511  mplsubrglem  17517  mplmonmul  17543  mplcoe1  17544  mplcoe5  17548  mplbas2  17551  evlslem4  17591  evlslem2  17597  uvcresum  18218  frlmsslsp  18223  rrxcph  20896  rrxmval  20904  rrxmetlem  20906  rrxmet  20907  rrxdstprj1  20908  deg1mul3le  21588  eulerpartlemb  26751