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Theorem ressuppssdif 6839
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 3399 . . . . . 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 3954 . . . . . . . . . 10  |-  ( z  =  x  ->  { z }  =  { x } )
32imaeq2d 5249 . . . . . . . . 9  |-  ( z  =  x  ->  ( F " { z } )  =  ( F
" { x }
) )
43neeq1d 2659 . . . . . . . 8  |-  ( z  =  x  ->  (
( F " {
z } )  =/= 
{ Z }  <->  ( F " { x } )  =/=  { Z }
) )
54elrab 3182 . . . . . . 7  |-  ( x  e.  { z  e. 
dom  F  |  ( F " { z } )  =/=  { Z } }  <->  ( x  e. 
dom  F  /\  ( F " { x }
)  =/=  { Z } ) )
6 ianor 486 . . . . . . . 8  |-  ( -.  ( x  e.  dom  ( F  |`  B )  /\  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } )  <-> 
( -.  x  e. 
dom  ( F  |`  B )  \/  -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } ) )
72imaeq2d 5249 . . . . . . . . . 10  |-  ( z  =  x  ->  (
( F  |`  B )
" { z } )  =  ( ( F  |`  B ) " { x } ) )
87neeq1d 2659 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ( F  |`  B ) " {
z } )  =/= 
{ Z }  <->  ( ( F  |`  B ) " { x } )  =/=  { Z }
) )
98elrab 3182 . . . . . . . 8  |-  ( x  e.  { z  e. 
dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } }  <->  ( x  e. 
dom  ( F  |`  B )  /\  (
( F  |`  B )
" { x }
)  =/=  { Z } ) )
106, 9xchnxbir 307 . . . . . . 7  |-  ( -.  x  e.  { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B )
" { z } )  =/=  { Z } }  <->  ( -.  x  e.  dom  ( F  |`  B )  \/  -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } ) )
11 ianor 486 . . . . . . . . . . 11  |-  ( -.  ( x  e.  B  /\  x  e.  dom  F )  <->  ( -.  x  e.  B  \/  -.  x  e.  dom  F ) )
12 dmres 5206 . . . . . . . . . . . 12  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
1312elin2 3603 . . . . . . . . . . 11  |-  ( x  e.  dom  ( F  |`  B )  <->  ( x  e.  B  /\  x  e.  dom  F ) )
1411, 13xchnxbir 307 . . . . . . . . . 10  |-  ( -.  x  e.  dom  ( F  |`  B )  <->  ( -.  x  e.  B  \/  -.  x  e.  dom  F ) )
15 simpl 455 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  x  e.  dom  F )
1615anim2i 567 . . . . . . . . . . . . . 14  |-  ( ( -.  x  e.  B  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  ( -.  x  e.  B  /\  x  e.  dom  F ) )
1716ancomd 449 . . . . . . . . . . . . 13  |-  ( ( -.  x  e.  B  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  (
x  e.  dom  F  /\  -.  x  e.  B
) )
18 eldif 3399 . . . . . . . . . . . . 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 432 . . . . . . . . . . 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 463 . . . . . . . . . . . 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 377 . . . . . . . . . 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 464 . . . . . . . . . . 11  |-  ( ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  x  e.  dom  F )
27 snssi 4088 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  B  ->  { x }  C_  B )
2827adantl 464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  { x }  C_  B )
29 resima2 5219 . . . . . . . . . . . . . . . . . . . 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 2390 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( F " { x } )  =  ( ( F  |`  B ) " {
x } ) )
3231adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( ( F  |`  B ) " { x } )  =  { Z }
)  ->  ( F " { x } )  =  ( ( F  |`  B ) " {
x } ) )
33 simpr 459 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( ( F  |`  B ) " { x } )  =  { Z }
)  ->  ( ( F  |`  B ) " { x } )  =  { Z }
)
3432, 33eqtrd 2423 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( ( F  |`  B ) " { x } )  =  { Z }
)  ->  ( F " { x } )  =  { Z }
)
3534ex 432 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( (
( F  |`  B )
" { x }
)  =  { Z }  ->  ( F " { x } )  =  { Z }
) )
3635necon3d 2606 . . . . . . . . . . . . . 14  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( ( F " { x }
)  =/=  { Z }  ->  ( ( F  |`  B ) " {
x } )  =/= 
{ Z } ) )
3736impancom 438 . . . . . . . . . . . . 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 428 . . . . . . . . . . 11  |-  ( ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  -.  x  e.  B )
4026, 39eldifd 3400 . . . . . . . . . 10  |-  ( ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  /\  ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
) )  ->  x  e.  ( dom  F  \  B ) )
4140ex 432 . . . . . . . . 9  |-  ( -.  ( ( F  |`  B ) " {
x } )  =/= 
{ Z }  ->  ( ( x  e.  dom  F  /\  ( F " { x } )  =/=  { Z }
)  ->  x  e.  ( dom  F  \  B
) ) )
4225, 41jaoi 377 . . . . . . . 8  |-  ( ( -.  x  e.  dom  ( F  |`  B )  \/  -.  ( ( F  |`  B ) " { x } )  =/=  { Z }
)  ->  ( (
x  e.  dom  F  /\  ( F " {
x } )  =/= 
{ Z } )  ->  x  e.  ( dom  F  \  B
) ) )
4342impcom 428 . . . . . . 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 477 . . . . . 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 3423 . . 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 3827 . . 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 6819 . 2  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { z  e. 
dom  F  |  ( F " { z } )  =/=  { Z } } )
51 resexg 5228 . . . 4  |-  ( F  e.  V  ->  ( F  |`  B )  e. 
_V )
52 suppval 6819 . . . 4  |-  ( ( ( F  |`  B )  e.  _V  /\  Z  e.  W )  ->  (
( F  |`  B ) supp 
Z )  =  {
z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
)
5351, 52sylan 469 . . 3  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( F  |`  B ) supp  Z )  =  { z  e.  dom  ( F  |`  B )  |  ( ( F  |`  B ) " {
z } )  =/= 
{ Z } }
)
5453uneq1d 3571 . 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 3460 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 366    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   {crab 2736   _Vcvv 3034    \ cdif 3386    u. cun 3387    C_ wss 3389   {csn 3944   dom cdm 4913    |` cres 4915   "cima 4916  (class class class)co 6196   supp csupp 6817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-supp 6818
This theorem is referenced by:  ressuppfi  7770
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