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Theorem ressuppssdif 6710
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
ressuppssdif  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z ) 
C_  ( ( ( F  |`  B ) supp  Z )  u.  ( dom 
F  \  B )
) )

Proof of Theorem ressuppssdif
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldif 3338 . . . . . 6  |-  ( x  e.  ( { z  e.  dom  F  | 
( F " {
z } )  =/= 
{ Z } }  \  { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
)  <->  ( x  e. 
{ z  e.  dom  F  |  ( F " { z } )  =/=  { Z } }  /\  -.  x  e. 
{ z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
) )
2 sneq 3887 . . . . . . . . . 10  |-  ( z  =  x  ->  { z }  =  { x } )
32imaeq2d 5169 . . . . . . . . 9  |-  ( z  =  x  ->  ( F " { z } )  =  ( F
" { x }
) )
43neeq1d 2621 . . . . . . . 8  |-  ( z  =  x  ->  (
( F " {
z } )  =/= 
{ Z }  <->  ( F " { x } )  =/=  { Z }
) )
54elrab 3117 . . . . . . 7  |-  ( x  e.  { z  e. 
dom  F  |  ( F " { z } )  =/=  { Z } }  <->  ( x  e. 
dom  F  /\  ( F " { x }
)  =/=  { Z } ) )
6 ianor 488 . . . . . . . 8  |-  ( -.  ( x  e.  dom  ( F  |`  B )  /\  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } )  <-> 
( -.  x  e. 
dom  ( F  |`  B )  \/  -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } ) )
72imaeq2d 5169 . . . . . . . . . 10  |-  ( z  =  x  ->  (
( F  |`  B )
" { z } )  =  ( ( F  |`  B ) " { x } ) )
87neeq1d 2621 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ( F  |`  B ) " {
z } )  =/= 
{ Z }  <->  ( ( F  |`  B ) " { x } )  =/=  { Z }
) )
98elrab 3117 . . . . . . . 8  |-  ( x  e.  { z  e. 
dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } }  <->  ( x  e. 
dom  ( F  |`  B )  /\  (
( F  |`  B )
" { x }
)  =/=  { Z } ) )
106, 9xchnxbir 309 . . . . . . 7  |-  ( -.  x  e.  { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } }  <->  ( -.  x  e.  dom  ( F  |`  B )  \/  -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } ) )
11 ianor 488 . . . . . . . . . . 11  |-  ( -.  ( x  e.  B  /\  x  e.  dom  F )  <->  ( -.  x  e.  B  \/  -.  x  e.  dom  F ) )
12 dmres 5131 . . . . . . . . . . . 12  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
1312elin2 3541 . . . . . . . . . . 11  |-  ( x  e.  dom  ( F  |`  B )  <->  ( x  e.  B  /\  x  e.  dom  F ) )
1411, 13xchnxbir 309 . . . . . . . . . 10  |-  ( -.  x  e.  dom  ( F  |`  B )  <->  ( -.  x  e.  B  \/  -.  x  e.  dom  F ) )
15 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  x  e.  dom  F )
1615anim2i 569 . . . . . . . . . . . . . 14  |-  ( ( -.  x  e.  B  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  ( -.  x  e.  B  /\  x  e.  dom  F ) )
1716ancomd 451 . . . . . . . . . . . . 13  |-  ( ( -.  x  e.  B  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  (
x  e.  dom  F  /\  -.  x  e.  B
) )
18 eldif 3338 . . . . . . . . . . . . 13  |-  ( x  e.  ( dom  F  \  B )  <->  ( x  e.  dom  F  /\  -.  x  e.  B )
)
1917, 18sylibr 212 . . . . . . . . . . . 12  |-  ( ( -.  x  e.  B  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  x  e.  ( dom  F  \  B ) )
2019ex 434 . . . . . . . . . . 11  |-  ( -.  x  e.  B  -> 
( ( x  e. 
dom  F  /\  ( F " { x }
)  =/=  { Z } )  ->  x  e.  ( dom  F  \  B ) ) )
21 pm2.24 109 . . . . . . . . . . . . 13  |-  ( x  e.  dom  F  -> 
( -.  x  e. 
dom  F  ->  x  e.  ( dom  F  \  B ) ) )
2221adantr 465 . . . . . . . . . . . 12  |-  ( ( x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  ( -.  x  e.  dom  F  ->  x  e.  ( dom  F  \  B ) ) )
2322com12 31 . . . . . . . . . . 11  |-  ( -.  x  e.  dom  F  ->  ( ( x  e. 
dom  F  /\  ( F " { x }
)  =/=  { Z } )  ->  x  e.  ( dom  F  \  B ) ) )
2420, 23jaoi 379 . . . . . . . . . 10  |-  ( ( -.  x  e.  B  \/  -.  x  e.  dom  F )  ->  ( (
x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  x  e.  ( dom  F  \  B
) ) )
2514, 24sylbi 195 . . . . . . . . 9  |-  ( -.  x  e.  dom  ( F  |`  B )  -> 
( ( x  e. 
dom  F  /\  ( F " { x }
)  =/=  { Z } )  ->  x  e.  ( dom  F  \  B ) ) )
2615adantl 466 . . . . . . . . . . 11  |-  ( ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  x  e.  dom  F )
27 snssi 4017 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  B  ->  { x }  C_  B )
2827adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  { x }  C_  B )
29 resima2 5143 . . . . . . . . . . . . . . . . . . . 20  |-  ( { x }  C_  B  ->  ( ( F  |`  B ) " {
x } )  =  ( F " {
x } ) )
3028, 29syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( ( F  |`  B ) " { x } )  =  ( F " { x } ) )
3130eqcomd 2448 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( F " { x } )  =  ( ( F  |`  B ) " {
x } ) )
3231adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( ( F  |`  B ) " { x } )  =  { Z }
)  ->  ( F " { x } )  =  ( ( F  |`  B ) " {
x } ) )
33 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( ( F  |`  B ) " { x } )  =  { Z }
)  ->  ( ( F  |`  B ) " { x } )  =  { Z }
)
3432, 33eqtrd 2475 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( ( F  |`  B ) " { x } )  =  { Z }
)  ->  ( F " { x } )  =  { Z }
)
3534ex 434 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( (
( F  |`  B )
" { x }
)  =  { Z }  ->  ( F " { x } )  =  { Z }
) )
3635necon3d 2646 . . . . . . . . . . . . . 14  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( ( F " { x }
)  =/=  { Z }  ->  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } ) )
3736impancom 440 . . . . . . . . . . . . 13  |-  ( ( x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  ( x  e.  B  ->  ( ( F  |`  B ) " { x } )  =/=  { Z }
) )
3837con3d 133 . . . . . . . . . . . 12  |-  ( ( x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  ( -.  (
( F  |`  B )
" { x }
)  =/=  { Z }  ->  -.  x  e.  B ) )
3938impcom 430 . . . . . . . . . . 11  |-  ( ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  -.  x  e.  B )
4026, 39eldifd 3339 . . . . . . . . . 10  |-  ( ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  x  e.  ( dom  F  \  B ) )
4140ex 434 . . . . . . . . 9  |-  ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  ->  ( ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
)  ->  x  e.  ( dom  F  \  B
) ) )
4225, 41jaoi 379 . . . . . . . 8  |-  ( ( -.  x  e.  dom  ( F  |`  B )  \/  -.  ( ( F  |`  B ) " { x } )  =/=  { Z }
)  ->  ( (
x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  x  e.  ( dom  F  \  B
) ) )
4342impcom 430 . . . . . . 7  |-  ( ( ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
)  /\  ( -.  x  e.  dom  ( F  |`  B )  \/  -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } ) )  ->  x  e.  ( dom  F  \  B
) )
445, 10, 43syl2anb 479 . . . . . 6  |-  ( ( x  e.  { z  e.  dom  F  | 
( F " {
z } )  =/= 
{ Z } }  /\  -.  x  e.  {
z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
)  ->  x  e.  ( dom  F  \  B
) )
451, 44sylbi 195 . . . . 5  |-  ( x  e.  ( { z  e.  dom  F  | 
( F " {
z } )  =/= 
{ Z } }  \  { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
)  ->  x  e.  ( dom  F  \  B
) )
4645a1i 11 . . . 4  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( x  e.  ( { z  e.  dom  F  |  ( F " { z } )  =/=  { Z } }  \  { z  e. 
dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } } )  ->  x  e.  ( dom  F  \  B ) ) )
4746ssrdv 3362 . . 3  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( { z  e. 
dom  F  |  ( F " { z } )  =/=  { Z } }  \  { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } } )  C_  ( dom  F  \  B ) )
48 ssundif 3762 . . 3  |-  ( { z  e.  dom  F  |  ( F " { z } )  =/=  { Z } }  C_  ( { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } }  u.  ( dom  F  \  B ) )  <->  ( { z  e.  dom  F  | 
( F " {
z } )  =/= 
{ Z } }  \  { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
)  C_  ( dom  F 
\  B ) )
4947, 48sylibr 212 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  { z  e.  dom  F  |  ( F " { z } )  =/=  { Z } }  C_  ( { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } }  u.  ( dom  F  \  B ) ) )
50 suppval 6692 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { z  e. 
dom  F  |  ( F " { z } )  =/=  { Z } } )
51 resexg 5149 . . . 4  |-  ( F  e.  V  ->  ( F  |`  B )  e. 
_V )
52 suppval 6692 . . . 4  |-  ( ( ( F  |`  B )  e.  _V  /\  Z  e.  W )  ->  (
( F  |`  B ) supp 
Z )  =  {
z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
)
5351, 52sylan 471 . . 3  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( F  |`  B ) supp  Z )  =  { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
)
5453uneq1d 3509 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( ( F  |`  B ) supp  Z )  u.  ( dom  F  \  B ) )  =  ( { z  e. 
dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } }  u.  ( dom  F  \  B ) ) )
5549, 50, 543sstr4d 3399 1  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z ) 
C_  ( ( ( F  |`  B ) supp  Z )  u.  ( dom 
F  \  B )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   {crab 2719   _Vcvv 2972    \ cdif 3325    u. cun 3326    C_ wss 3328   {csn 3877   dom cdm 4840    |` cres 4842   "cima 4843  (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-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-rab 2724  df-v 2974  df-sbc 3187  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-br 4293  df-opab 4351  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-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-supp 6691
This theorem is referenced by:  ressuppfi  7646
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