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Theorem ressuppssdif 6921
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 3486 . . . . . 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 4037 . . . . . . . . . 10  |-  ( z  =  x  ->  { z }  =  { x } )
32imaeq2d 5336 . . . . . . . . 9  |-  ( z  =  x  ->  ( F " { z } )  =  ( F
" { x }
) )
43neeq1d 2744 . . . . . . . 8  |-  ( z  =  x  ->  (
( F " {
z } )  =/= 
{ Z }  <->  ( F " { x } )  =/=  { Z }
) )
54elrab 3261 . . . . . . 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 5336 . . . . . . . . . 10  |-  ( z  =  x  ->  (
( F  |`  B )
" { z } )  =  ( ( F  |`  B ) " { x } ) )
87neeq1d 2744 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ( F  |`  B ) " {
z } )  =/= 
{ Z }  <->  ( ( F  |`  B ) " { x } )  =/=  { Z }
) )
98elrab 3261 . . . . . . . 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 5293 . . . . . . . . . . . 12  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
1312elin2 3689 . . . . . . . . . . 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 3486 . . . . . . . . . . . . 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 4171 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  B  ->  { x }  C_  B )
2827adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  { x }  C_  B )
29 resima2 5306 . . . . . . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . . . . . 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 2508 . . . . . . . . . . . . . . . 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 2691 . . . . . . . . . . . . . 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 3487 . . . . . . . . . 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 3510 . . 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 3910 . . 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 6903 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { z  e. 
dom  F  |  ( F " { z } )  =/=  { Z } } )
51 resexg 5315 . . . 4  |-  ( F  e.  V  ->  ( F  |`  B )  e. 
_V )
52 suppval 6903 . . . 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 3657 . 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 3547 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 1379    e. wcel 1767    =/= wne 2662   {crab 2818   _Vcvv 3113    \ cdif 3473    u. cun 3474    C_ wss 3476   {csn 4027   dom cdm 4999    |` cres 5001   "cima 5002  (class class class)co 6283   supp csupp 6901
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 4568  ax-nul 4576  ax-pr 4686  ax-un 6575
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-supp 6902
This theorem is referenced by:  ressuppfi  7854
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