Theorem suppssr 6957

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 3452	. 2 |- ( X e. ( A \ W ) <-> ( X e. A /\ -. X e. W ) )
2		fvex 5891	. . . . . 6 |- ( F ` X ) e. _V
3		eldifsn 4128	. . . . . 6 |- ( ( F ` X ) e. ( _V \ { Z } ) <-> ( ( F ` X ) e. _V /\ ( F ` X ) =/= Z ) )
4	2, 3	mpbiran 926	. . . . 5 |- ( ( F ` X ) e. ( _V \ { Z } ) <-> ( F ` X ) =/= Z )
5		suppssr.f	. . . . . . . . . 10 |- ( ph -> F : A --> B )
6		ffn 5746	. . . . . . . . . 10 |- ( F : A --> B -> F Fn A )
7	5, 6	syl 17	. . . . . . . . 9 |- ( ph -> F Fn A )
8		suppssr.a	. . . . . . . . 9 |- ( ph -> A e. V )
9		suppssr.z	. . . . . . . . 9 |- ( ph -> Z e. U )
10		elsuppfn 6933	. . . . . . . . 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 1264	. . . . . . . 8 |- ( ph -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) =/= Z ) ) )
12		ibar 506	. . . . . . . . . . 11 |- ( ( F ` X ) e. _V -> ( ( F ` X ) =/= Z <-> ( ( F ` X ) e. _V /\ ( F ` X ) =/= Z ) ) )
13	2, 12	mp1i 13	. . . . . . . . . 10 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z <-> ( ( F ` X ) e. _V /\ ( F ` X ) =/= Z ) ) )
14	13, 3	syl6bbr 266	. . . . . . . . 9 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z <-> ( F ` X ) e. ( _V \ { Z } ) ) )
15	14	pm5.32da 645	. . . . . . . 8 |- ( ph -> ( ( X e. A /\ ( F ` X ) =/= Z ) <-> ( X e. A /\ ( F ` X ) e. ( _V \ { Z } ) ) ) )
16	11, 15	bitrd 256	. . . . . . 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 3469	. . . . . . 7 |- ( ph -> ( X e. ( F supp Z ) -> X e. W ) )
19	16, 18	sylbird 238	. . . . . 6 |- ( ph -> ( ( X e. A /\ ( F ` X ) e. ( _V \ { Z } ) ) -> X e. W ) )
20	19	expdimp 438	. . . . 5 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) e. ( _V \ { Z } ) -> X e. W ) )
21	4, 20	syl5bir 221	. . . 4 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z -> X e. W ) )
22	21	necon1bd 2649	. . 3 |- ( ( ph /\ X e. A ) -> ( -. X e. W -> ( F ` X ) = Z ) )
23	22	impr 623	. 2 |- ( ( ph /\ ( X e. A /\ -. X e. W ) ) -> ( F ` X ) = Z )
24	1, 23	sylan2b 477	1 |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1870   =/= wne 2625   _Vcvv 3087   \ cdif 3439   C_ wss 3442   {csn 4002   Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   supp csupp 6925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-supp 6926

This theorem is referenced by:  fsuppmptif  7919  fsuppco2  7922  fsuppcor  7923  cantnfp1lem1  8182  cantnfp1lem3  8184  cantnflem1  8193  cnfcom2lem  8205  gsumval3  17476  gsumcllem  17477  gsumzaddlem  17489  gsumzmhm  17505  gsumpt  17529  gsum2dlem1  17537  gsum2dlem2  17538  gsum2d  17539  dprdfinv  17587  dprdfadd  17588  dmdprdsplitlem  17605  dpjidcl  17626  gsumdixp  17772  lcomfsupp  18063  psrbaglesupp  18527  psrbagaddcl  18529  psrbaglefi  18531  mplsubglem  18593  mpllsslem  18594  mplsubrglem  18598  mplmonmul  18623  mplcoe1  18624  mplcoe5  18627  mplbas2  18629  evlslem4  18666  evlslem2  18670  uvcresum  19282  frlmsslsp  19285  rrxcph  22244  rrxmval  22252  rrxmetlem  22254  rrxmet  22255  rrxdstprj1  22256  deg1mul3le  22942  eulerpartlemb  29027