MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptsuppdifd Structured version   Unicode version

Theorem mptsuppdifd 6813
Description: The support of a function in maps to notation with a set difference. (Contributed by AV, 28-May-2019.)
Hypotheses
Ref Expression
mptsuppdifd.f  |-  F  =  ( x  e.  A  |->  B )
mptsuppdifd.a  |-  ( ph  ->  A  e.  V )
mptsuppdifd.z  |-  ( ph  ->  Z  e.  W )
Assertion
Ref Expression
mptsuppdifd  |-  ( ph  ->  ( F supp  Z )  =  { x  e.  A  |  B  e.  ( _V  \  { Z } ) } )
Distinct variable groups:    x, A    x, Z
Allowed substitution hints:    ph( x)    B( x)    F( x)    V( x)    W( x)

Proof of Theorem mptsuppdifd
StepHypRef Expression
1 mptsuppdifd.f . . . 4  |-  F  =  ( x  e.  A  |->  B )
2 mptsuppdifd.a . . . . 5  |-  ( ph  ->  A  e.  V )
3 mptexg 6048 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
42, 3syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  _V )
51, 4syl5eqel 2543 . . 3  |-  ( ph  ->  F  e.  _V )
6 mptsuppdifd.z . . 3  |-  ( ph  ->  Z  e.  W )
7 suppimacnv 6803 . . 3  |-  ( ( F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
85, 6, 7syl2anc 661 . 2  |-  ( ph  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
91mptpreima 5431 . 2  |-  ( `' F " ( _V 
\  { Z }
) )  =  {
x  e.  A  |  B  e.  ( _V  \  { Z } ) }
108, 9syl6eq 2508 1  |-  ( ph  ->  ( F supp  Z )  =  { x  e.  A  |  B  e.  ( _V  \  { Z } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   {crab 2799   _Vcvv 3070    \ cdif 3425   {csn 3977    |-> cmpt 4450   `'ccnv 4939   "cima 4943  (class class class)co 6192   supp csupp 6792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-supp 6793
This theorem is referenced by:  mptsuppd  6814  extmptsuppeq  6815  suppssov1  6823  suppss2  6825  suppssfv  6827
  Copyright terms: Public domain W3C validator