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Theorem ressuppss 6931
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 3692 . . . . . . . . 9  |-  ( b  e.  ( B  i^i  dom 
F )  <->  ( b  e.  B  /\  b  e.  dom  F ) )
21simprbi 464 . . . . . . . 8  |-  ( b  e.  ( B  i^i  dom 
F )  ->  b  e.  dom  F )
3 dmres 5300 . . . . . . . 8  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
42, 3eleq2s 2575 . . . . . . 7  |-  ( b  e.  dom  ( F  |`  B )  ->  b  e.  dom  F )
54ad2antrl 727 . . . . . 6  |-  ( ( ( F  e.  V  /\  Z  e.  W
)  /\  ( b  e.  dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } ) )  -> 
b  e.  dom  F
)
6 snssi 4177 . . . . . . . . . . . 12  |-  ( b  e.  B  ->  { b }  C_  B )
7 resima2 5313 . . . . . . . . . . . 12  |-  ( { b }  C_  B  ->  ( ( F  |`  B ) " {
b } )  =  ( F " {
b } ) )
86, 7syl 16 . . . . . . . . . . 11  |-  ( b  e.  B  ->  (
( F  |`  B )
" { b } )  =  ( F
" { b } ) )
98neeq1d 2744 . . . . . . . . . 10  |-  ( b  e.  B  ->  (
( ( F  |`  B ) " {
b } )  =/= 
{ Z }  <->  ( F " { b } )  =/=  { Z }
) )
109biimpd 207 . . . . . . . . 9  |-  ( b  e.  B  ->  (
( ( F  |`  B ) " {
b } )  =/= 
{ Z }  ->  ( F " { b } )  =/=  { Z } ) )
1110adantld 467 . . . . . . . 8  |-  ( b  e.  B  ->  (
( b  e.  dom  ( F  |`  B )  /\  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } )  ->  ( F " { b } )  =/=  { Z }
) )
1211adantld 467 . . . . . . 7  |-  ( b  e.  B  ->  (
( ( F  e.  V  /\  Z  e.  W )  /\  (
b  e.  dom  ( F  |`  B )  /\  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } ) )  ->  ( F " { b } )  =/=  { Z }
) )
13 pm2.24 109 . . . . . . . . . . . 12  |-  ( b  e.  B  ->  ( -.  b  e.  B  ->  ( F " {
b } )  =/= 
{ Z } ) )
1413adantr 465 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  b  e.  dom  F )  ->  ( -.  b  e.  B  ->  ( F
" { b } )  =/=  { Z } ) )
151, 14sylbi 195 . . . . . . . . . 10  |-  ( b  e.  ( B  i^i  dom 
F )  ->  ( -.  b  e.  B  ->  ( F " {
b } )  =/= 
{ Z } ) )
1615, 3eleq2s 2575 . . . . . . . . 9  |-  ( b  e.  dom  ( F  |`  B )  ->  ( -.  b  e.  B  ->  ( F " {
b } )  =/= 
{ Z } ) )
1716ad2antrl 727 . . . . . . . 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 164 . . . . . 6  |-  ( ( ( F  e.  V  /\  Z  e.  W
)  /\  ( b  e.  dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } ) )  -> 
( F " {
b } )  =/= 
{ Z } )
205, 19jca 532 . . . . 5  |-  ( ( ( F  e.  V  /\  Z  e.  W
)  /\  ( b  e.  dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } ) )  -> 
( b  e.  dom  F  /\  ( F " { b } )  =/=  { Z }
) )
2120ex 434 . . . 4  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( b  e. 
dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } )  ->  (
b  e.  dom  F  /\  ( F " {
b } )  =/= 
{ Z } ) ) )
2221ss2abdv 3578 . . 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 2826 . . 3  |-  { b  e.  dom  ( F  |`  B )  |  ( ( F  |`  B )
" { b } )  =/=  { Z } }  =  {
b  |  ( b  e.  dom  ( F  |`  B )  /\  (
( F  |`  B )
" { b } )  =/=  { Z } ) }
24 df-rab 2826 . . 3  |-  { b  e.  dom  F  | 
( F " {
b } )  =/= 
{ Z } }  =  { b  |  ( b  e.  dom  F  /\  ( F " {
b } )  =/= 
{ Z } ) }
2522, 23, 243sstr4g 3550 . 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 5322 . . 3  |-  ( F  e.  V  ->  ( F  |`  B )  e. 
_V )
27 suppval 6915 . . 3  |-  ( ( ( F  |`  B )  e.  _V  /\  Z  e.  W )  ->  (
( F  |`  B ) supp 
Z )  =  {
b  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } }
)
2826, 27sylan 471 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( F  |`  B ) supp  Z )  =  { b  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
b } )  =/= 
{ Z } }
)
29 suppval 6915 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { b  e. 
dom  F  |  ( F " { b } )  =/=  { Z } } )
3025, 28, 293sstr4d 3552 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 369    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   {crab 2821   _Vcvv 3118    i^i cin 3480    C_ wss 3481   {csn 4033   dom cdm 5005    |` cres 5007   "cima 5008  (class class class)co 6295   supp csupp 6913
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-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-supp 6914
This theorem is referenced by:  fsuppres  7866  gsumzres  16787  gsumzadd  16808  gsum2dlem2  16871  tsmsres  20514
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