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Theorem mptsuppdifd 6920
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 6123 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
42, 3syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  _V )
51, 4syl5eqel 2533 . . 3  |-  ( ph  ->  F  e.  _V )
6 mptsuppdifd.z . . 3  |-  ( ph  ->  Z  e.  W )
7 suppimacnv 6910 . . 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 5486 . 2  |-  ( `' F " ( _V 
\  { Z }
) )  =  {
x  e.  A  |  B  e.  ( _V  \  { Z } ) }
108, 9syl6eq 2498 1  |-  ( ph  ->  ( F supp  Z )  =  { x  e.  A  |  B  e.  ( _V  \  { Z } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802   {crab 2795   _Vcvv 3093    \ cdif 3455   {csn 4010    |-> cmpt 4491   `'ccnv 4984   "cima 4988  (class class class)co 6277   supp csupp 6899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-supp 6900
This theorem is referenced by:  mptsuppd  6921  extmptsuppeq  6922  suppssov1  6930  suppss2  6932  suppssfv  6934
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