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Theorem mptsuppd 6926
Description: The support of a function in maps to notation. (Contributed by AV, 10-Apr-2019.) (Revised 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 )
mptsuppd.b  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  U )
Assertion
Ref Expression
mptsuppd  |-  ( ph  ->  ( F supp  Z )  =  { x  e.  A  |  B  =/= 
Z } )
Distinct variable groups:    x, A    x, Z    ph, x
Allowed substitution hints:    B( x)    U( x)    F( x)    V( x)    W( x)

Proof of Theorem mptsuppd
StepHypRef Expression
1 mptsuppdifd.f . . 3  |-  F  =  ( x  e.  A  |->  B )
2 mptsuppdifd.a . . 3  |-  ( ph  ->  A  e.  V )
3 mptsuppdifd.z . . 3  |-  ( ph  ->  Z  e.  W )
41, 2, 3mptsuppdifd 6925 . 2  |-  ( ph  ->  ( F supp  Z )  =  { x  e.  A  |  B  e.  ( _V  \  { Z } ) } )
5 mptsuppd.b . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  U )
6 elex 3068 . . . . . 6  |-  ( B  e.  U  ->  B  e.  _V )
75, 6syl 17 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  _V )
87biantrurd 506 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =/=  Z  <->  ( B  e.  _V  /\  B  =/= 
Z ) ) )
9 eldifsn 4097 . . . 4  |-  ( B  e.  ( _V  \  { Z } )  <->  ( B  e.  _V  /\  B  =/= 
Z ) )
108, 9syl6rbbr 264 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( B  e.  ( _V  \  { Z } )  <-> 
B  =/=  Z ) )
1110rabbidva 3050 . 2  |-  ( ph  ->  { x  e.  A  |  B  e.  ( _V  \  { Z }
) }  =  {
x  e.  A  |  B  =/=  Z } )
124, 11eqtrd 2443 1  |-  ( ph  ->  ( F supp  Z )  =  { x  e.  A  |  B  =/= 
Z } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   {crab 2758   _Vcvv 3059    \ cdif 3411   {csn 3972    |-> cmpt 4453  (class class class)co 6278   supp csupp 6902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-supp 6903
This theorem is referenced by:  rmsupp0  38472  domnmsuppn0  38473  rmsuppss  38474  suppmptcfin  38483  lcoc0  38534  linc1  38537
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