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Theorem ressuppssdif 6945
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 3447 . . . . . 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 4007 . . . . . . . . . 10  |-  ( z  =  x  ->  { z }  =  { x } )
32imaeq2d 5185 . . . . . . . . 9  |-  ( z  =  x  ->  ( F " { z } )  =  ( F
" { x }
) )
43neeq1d 2702 . . . . . . . 8  |-  ( z  =  x  ->  (
( F " {
z } )  =/= 
{ Z }  <->  ( F " { x } )  =/=  { Z }
) )
54elrab 3230 . . . . . . 7  |-  ( x  e.  { z  e. 
dom  F  |  ( F " { z } )  =/=  { Z } }  <->  ( x  e. 
dom  F  /\  ( F " { x }
)  =/=  { Z } ) )
6 ianor 491 . . . . . . . 8  |-  ( -.  ( x  e.  dom  ( F  |`  B )  /\  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } )  <-> 
( -.  x  e. 
dom  ( F  |`  B )  \/  -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } ) )
72imaeq2d 5185 . . . . . . . . . 10  |-  ( z  =  x  ->  (
( F  |`  B )
" { z } )  =  ( ( F  |`  B ) " { x } ) )
87neeq1d 2702 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ( F  |`  B ) " {
z } )  =/= 
{ Z }  <->  ( ( F  |`  B ) " { x } )  =/=  { Z }
) )
98elrab 3230 . . . . . . . 8  |-  ( x  e.  { z  e. 
dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } }  <->  ( x  e. 
dom  ( F  |`  B )  /\  (
( F  |`  B )
" { x }
)  =/=  { Z } ) )
106, 9xchnxbir 311 . . . . . . 7  |-  ( -.  x  e.  { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } }  <->  ( -.  x  e.  dom  ( F  |`  B )  \/  -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } ) )
11 ianor 491 . . . . . . . . . . 11  |-  ( -.  ( x  e.  B  /\  x  e.  dom  F )  <->  ( -.  x  e.  B  \/  -.  x  e.  dom  F ) )
12 dmres 5142 . . . . . . . . . . . 12  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
1312elin2 3654 . . . . . . . . . . 11  |-  ( x  e.  dom  ( F  |`  B )  <->  ( x  e.  B  /\  x  e.  dom  F ) )
1411, 13xchnxbir 311 . . . . . . . . . 10  |-  ( -.  x  e.  dom  ( F  |`  B )  <->  ( -.  x  e.  B  \/  -.  x  e.  dom  F ) )
15 simpl 459 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  x  e.  dom  F )
1615anim2i 572 . . . . . . . . . . . . . 14  |-  ( ( -.  x  e.  B  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  ( -.  x  e.  B  /\  x  e.  dom  F ) )
1716ancomd 453 . . . . . . . . . . . . 13  |-  ( ( -.  x  e.  B  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  (
x  e.  dom  F  /\  -.  x  e.  B
) )
18 eldif 3447 . . . . . . . . . . . . 13  |-  ( x  e.  ( dom  F  \  B )  <->  ( x  e.  dom  F  /\  -.  x  e.  B )
)
1917, 18sylibr 216 . . . . . . . . . . . 12  |-  ( ( -.  x  e.  B  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  x  e.  ( dom  F  \  B ) )
2019ex 436 . . . . . . . . . . 11  |-  ( -.  x  e.  B  -> 
( ( x  e. 
dom  F  /\  ( F " { x }
)  =/=  { Z } )  ->  x  e.  ( dom  F  \  B ) ) )
21 pm2.24 113 . . . . . . . . . . . . 13  |-  ( x  e.  dom  F  -> 
( -.  x  e. 
dom  F  ->  x  e.  ( dom  F  \  B ) ) )
2221adantr 467 . . . . . . . . . . . 12  |-  ( ( x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  ( -.  x  e.  dom  F  ->  x  e.  ( dom  F  \  B ) ) )
2322com12 33 . . . . . . . . . . 11  |-  ( -.  x  e.  dom  F  ->  ( ( x  e. 
dom  F  /\  ( F " { x }
)  =/=  { Z } )  ->  x  e.  ( dom  F  \  B ) ) )
2420, 23jaoi 381 . . . . . . . . . 10  |-  ( ( -.  x  e.  B  \/  -.  x  e.  dom  F )  ->  ( (
x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  x  e.  ( dom  F  \  B
) ) )
2514, 24sylbi 199 . . . . . . . . 9  |-  ( -.  x  e.  dom  ( F  |`  B )  -> 
( ( x  e. 
dom  F  /\  ( F " { x }
)  =/=  { Z } )  ->  x  e.  ( dom  F  \  B ) ) )
2615adantl 468 . . . . . . . . . . 11  |-  ( ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  x  e.  dom  F )
27 snssi 4142 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  B  ->  { x }  C_  B )
2827adantl 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  { x }  C_  B )
29 resima2 5155 . . . . . . . . . . . . . . . . . . . 20  |-  ( { x }  C_  B  ->  ( ( F  |`  B ) " {
x } )  =  ( F " {
x } ) )
3028, 29syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( ( F  |`  B ) " { x } )  =  ( F " { x } ) )
3130eqcomd 2431 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( F " { x } )  =  ( ( F  |`  B ) " {
x } ) )
3231adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( ( F  |`  B ) " { x } )  =  { Z }
)  ->  ( F " { x } )  =  ( ( F  |`  B ) " {
x } ) )
33 simpr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( ( F  |`  B ) " { x } )  =  { Z }
)  ->  ( ( F  |`  B ) " { x } )  =  { Z }
)
3432, 33eqtrd 2464 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( ( F  |`  B ) " { x } )  =  { Z }
)  ->  ( F " { x } )  =  { Z }
)
3534ex 436 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( (
( F  |`  B )
" { x }
)  =  { Z }  ->  ( F " { x } )  =  { Z }
) )
3635necon3d 2649 . . . . . . . . . . . . . 14  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( ( F " { x }
)  =/=  { Z }  ->  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } ) )
3736impancom 442 . . . . . . . . . . . . 13  |-  ( ( x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  ( x  e.  B  ->  ( ( F  |`  B ) " { x } )  =/=  { Z }
) )
3837con3d 139 . . . . . . . . . . . 12  |-  ( ( x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  ( -.  (
( F  |`  B )
" { x }
)  =/=  { Z }  ->  -.  x  e.  B ) )
3938impcom 432 . . . . . . . . . . 11  |-  ( ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  -.  x  e.  B )
4026, 39eldifd 3448 . . . . . . . . . 10  |-  ( ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  x  e.  ( dom  F  \  B ) )
4140ex 436 . . . . . . . . 9  |-  ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  ->  ( ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
)  ->  x  e.  ( dom  F  \  B
) ) )
4225, 41jaoi 381 . . . . . . . 8  |-  ( ( -.  x  e.  dom  ( F  |`  B )  \/  -.  ( ( F  |`  B ) " { x } )  =/=  { Z }
)  ->  ( (
x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  x  e.  ( dom  F  \  B
) ) )
4342impcom 432 . . . . . . 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 482 . . . . . 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 199 . . . . 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 3471 . . 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 3880 . . 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 216 . 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 6925 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { z  e. 
dom  F  |  ( F " { z } )  =/=  { Z } } )
51 resexg 5164 . . . 4  |-  ( F  e.  V  ->  ( F  |`  B )  e. 
_V )
52 suppval 6925 . . . 4  |-  ( ( ( F  |`  B )  e.  _V  /\  Z  e.  W )  ->  (
( F  |`  B ) supp 
Z )  =  {
z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
)
5351, 52sylan 474 . . 3  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( F  |`  B ) supp  Z )  =  { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
)
5453uneq1d 3620 . 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 3508 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 370    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   {crab 2780   _Vcvv 3082    \ cdif 3434    u. cun 3435    C_ wss 3437   {csn 3997   dom cdm 4851    |` cres 4853   "cima 4854  (class class class)co 6303   supp csupp 6923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-supp 6924
This theorem is referenced by:  ressuppfi  7913
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