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Theorem mptsuppdifd 6912
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 6121 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
42, 3syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  _V )
51, 4syl5eqel 2552 . . 3  |-  ( ph  ->  F  e.  _V )
6 mptsuppdifd.z . . 3  |-  ( ph  ->  Z  e.  W )
7 suppimacnv 6902 . . 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 5491 . 2  |-  ( `' F " ( _V 
\  { Z }
) )  =  {
x  e.  A  |  B  e.  ( _V  \  { Z } ) }
108, 9syl6eq 2517 1  |-  ( ph  ->  ( F supp  Z )  =  { x  e.  A  |  B  e.  ( _V  \  { Z } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   {crab 2811   _Vcvv 3106    \ cdif 3466   {csn 4020    |-> cmpt 4498   `'ccnv 4991   "cima 4995  (class class class)co 6275   supp csupp 6891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-supp 6892
This theorem is referenced by:  mptsuppd  6913  extmptsuppeq  6914  suppssov1  6922  suppss2  6924  suppssfv  6926
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