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Theorem ressuppss 6953
Description: The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
Assertion
Ref Expression
ressuppss  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( F  |`  B ) supp  Z )  C_  ( F supp  Z ) )

Proof of Theorem ressuppss
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 elin 3608 . . . . . . . . 9  |-  ( b  e.  ( B  i^i  dom 
F )  <->  ( b  e.  B  /\  b  e.  dom  F ) )
21simprbi 471 . . . . . . . 8  |-  ( b  e.  ( B  i^i  dom 
F )  ->  b  e.  dom  F )
3 dmres 5131 . . . . . . . 8  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
42, 3eleq2s 2567 . . . . . . 7  |-  ( b  e.  dom  ( F  |`  B )  ->  b  e.  dom  F )
54ad2antrl 742 . . . . . 6  |-  ( ( ( F  e.  V  /\  Z  e.  W
)  /\  ( b  e.  dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } ) )  -> 
b  e.  dom  F
)
6 snssi 4107 . . . . . . . . . . . 12  |-  ( b  e.  B  ->  { b }  C_  B )
7 resima2 5144 . . . . . . . . . . . 12  |-  ( { b }  C_  B  ->  ( ( F  |`  B ) " {
b } )  =  ( F " {
b } ) )
86, 7syl 17 . . . . . . . . . . 11  |-  ( b  e.  B  ->  (
( F  |`  B )
" { b } )  =  ( F
" { b } ) )
98neeq1d 2702 . . . . . . . . . 10  |-  ( b  e.  B  ->  (
( ( F  |`  B ) " {
b } )  =/= 
{ Z }  <->  ( F " { b } )  =/=  { Z }
) )
109biimpd 212 . . . . . . . . 9  |-  ( b  e.  B  ->  (
( ( F  |`  B ) " {
b } )  =/= 
{ Z }  ->  ( F " { b } )  =/=  { Z } ) )
1110adantld 474 . . . . . . . 8  |-  ( b  e.  B  ->  (
( b  e.  dom  ( F  |`  B )  /\  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } )  ->  ( F " { b } )  =/=  { Z }
) )
1211adantld 474 . . . . . . 7  |-  ( b  e.  B  ->  (
( ( F  e.  V  /\  Z  e.  W )  /\  (
b  e.  dom  ( F  |`  B )  /\  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } ) )  ->  ( F " { b } )  =/=  { Z }
) )
13 pm2.24 112 . . . . . . . . . . . 12  |-  ( b  e.  B  ->  ( -.  b  e.  B  ->  ( F " {
b } )  =/= 
{ Z } ) )
1413adantr 472 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  b  e.  dom  F )  ->  ( -.  b  e.  B  ->  ( F
" { b } )  =/=  { Z } ) )
151, 14sylbi 200 . . . . . . . . . 10  |-  ( b  e.  ( B  i^i  dom 
F )  ->  ( -.  b  e.  B  ->  ( F " {
b } )  =/= 
{ Z } ) )
1615, 3eleq2s 2567 . . . . . . . . 9  |-  ( b  e.  dom  ( F  |`  B )  ->  ( -.  b  e.  B  ->  ( F " {
b } )  =/= 
{ Z } ) )
1716ad2antrl 742 . . . . . . . 8  |-  ( ( ( F  e.  V  /\  Z  e.  W
)  /\  ( b  e.  dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } ) )  -> 
( -.  b  e.  B  ->  ( F " { b } )  =/=  { Z }
) )
1817com12 31 . . . . . . 7  |-  ( -.  b  e.  B  -> 
( ( ( F  e.  V  /\  Z  e.  W )  /\  (
b  e.  dom  ( F  |`  B )  /\  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } ) )  ->  ( F " { b } )  =/=  { Z }
) )
1912, 18pm2.61i 169 . . . . . 6  |-  ( ( ( F  e.  V  /\  Z  e.  W
)  /\  ( b  e.  dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } ) )  -> 
( F " {
b } )  =/= 
{ Z } )
205, 19jca 541 . . . . 5  |-  ( ( ( F  e.  V  /\  Z  e.  W
)  /\  ( b  e.  dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } ) )  -> 
( b  e.  dom  F  /\  ( F " { b } )  =/=  { Z }
) )
2120ex 441 . . . 4  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( b  e. 
dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } )  ->  (
b  e.  dom  F  /\  ( F " {
b } )  =/= 
{ Z } ) ) )
2221ss2abdv 3488 . . 3  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  { b  |  ( b  e.  dom  ( F  |`  B )  /\  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } ) }  C_  { b  |  ( b  e. 
dom  F  /\  ( F " { b } )  =/=  { Z } ) } )
23 df-rab 2765 . . 3  |-  { b  e.  dom  ( F  |`  B )  |  ( ( F  |`  B )
" { b } )  =/=  { Z } }  =  {
b  |  ( b  e.  dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } ) }
24 df-rab 2765 . . 3  |-  { b  e.  dom  F  | 
( F " {
b } )  =/= 
{ Z } }  =  { b  |  ( b  e.  dom  F  /\  ( F " {
b } )  =/= 
{ Z } ) }
2522, 23, 243sstr4g 3459 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  { b  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } }  C_ 
{ b  e.  dom  F  |  ( F " { b } )  =/=  { Z } } )
26 resexg 5153 . . 3  |-  ( F  e.  V  ->  ( F  |`  B )  e. 
_V )
27 suppval 6935 . . 3  |-  ( ( ( F  |`  B )  e.  _V  /\  Z  e.  W )  ->  (
( F  |`  B ) supp 
Z )  =  {
b  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } }
)
2826, 27sylan 479 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( F  |`  B ) supp  Z )  =  { b  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } }
)
29 suppval 6935 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { b  e. 
dom  F  |  ( F " { b } )  =/=  { Z } } )
3025, 28, 293sstr4d 3461 1  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( F  |`  B ) supp  Z )  C_  ( F supp  Z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457    =/= wne 2641   {crab 2760   _Vcvv 3031    i^i cin 3389    C_ wss 3390   {csn 3959   dom cdm 4839    |` cres 4841   "cima 4842  (class class class)co 6308   supp csupp 6933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-supp 6934
This theorem is referenced by:  fsuppres  7926  gsumzres  17621  gsumzadd  17633  gsum2dlem2  17681  tsmsres  21236
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