Theorem suppssr 6973

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 3426	. 2 |- ( X e. ( A \ W ) <-> ( X e. A /\ -. X e. W ) )
2		fvex 5898	. . . . . 6 |- ( F ` X ) e. _V
3		eldifsn 4110	. . . . . 6 |- ( ( F ` X ) e. ( _V \ { Z } ) <-> ( ( F ` X ) e. _V /\ ( F ` X ) =/= Z ) )
4	2, 3	mpbiran 934	. . . . 5 |- ( ( F ` X ) e. ( _V \ { Z } ) <-> ( F ` X ) =/= Z )
5		suppssr.f	. . . . . . . . . 10 |- ( ph -> F : A --> B )
6		ffn 5751	. . . . . . . . . 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 6949	. . . . . . . . 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 1276	. . . . . . . 8 |- ( ph -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) =/= Z ) ) )
12		ibar 511	. . . . . . . . . . 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 271	. . . . . . . . 9 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z <-> ( F ` X ) e. ( _V \ { Z } ) ) )
15	14	pm5.32da 651	. . . . . . . 8 |- ( ph -> ( ( X e. A /\ ( F ` X ) =/= Z ) <-> ( X e. A /\ ( F ` X ) e. ( _V \ { Z } ) ) ) )
16	11, 15	bitrd 261	. . . . . . 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 3443	. . . . . . 7 |- ( ph -> ( X e. ( F supp Z ) -> X e. W ) )
19	16, 18	sylbird 243	. . . . . 6 |- ( ph -> ( ( X e. A /\ ( F ` X ) e. ( _V \ { Z } ) ) -> X e. W ) )
20	19	expdimp 443	. . . . 5 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) e. ( _V \ { Z } ) -> X e. W ) )
21	4, 20	syl5bir 226	. . . 4 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z -> X e. W ) )
22	21	necon1bd 2654	. . 3 |- ( ( ph /\ X e. A ) -> ( -. X e. W -> ( F ` X ) = Z ) )
23	22	impr 629	. 2 |- ( ( ph /\ ( X e. A /\ -. X e. W ) ) -> ( F ` X ) = Z )
24	1, 23	sylan2b 482	1 |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   _Vcvv 3057   \ cdif 3413   C_ wss 3416   {csn 3980   Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6315   supp csupp 6941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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 6318  df-oprab 6319  df-mpt2 6320  df-supp 6942

This theorem is referenced by:  fsuppmptif  7939  fsuppco2  7942  fsuppcor  7943  cantnfp1lem1  8209  cantnfp1lem3  8211  cantnflem1  8220  cnfcom2lem  8232  gsumval3  17590  gsumcllem  17591  gsumzaddlem  17603  gsumzmhm  17619  gsumpt  17643  gsum2dlem1  17651  gsum2dlem2  17652  gsum2d  17653  dprdfinv  17701  dprdfadd  17702  dmdprdsplitlem  17719  dpjidcl  17740  gsumdixp  17886  lcomfsupp  18177  psrbaglesupp  18641  psrbagaddcl  18643  psrbaglefi  18645  mplsubglem  18707  mpllsslem  18708  mplsubrglem  18712  mplmonmul  18737  mplcoe1  18738  mplcoe5  18741  mplbas2  18743  evlslem4  18780  evlslem2  18784  uvcresum  19400  frlmsslsp  19403  rrxcph  22400  rrxmval  22408  rrxmetlem  22410  rrxmet  22411  rrxdstprj1  22412  deg1mul3le  23114  eulerpartlemb  29250