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Theorem mptsuppd 6823
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 6822 . 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 3087 . . . . . 6  |-  ( B  e.  U  ->  B  e.  _V )
75, 6syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  _V )
87biantrurd 508 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =/=  Z  <->  ( B  e.  _V  /\  B  =/= 
Z ) ) )
9 eldifsn 4109 . . . 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 3069 . 2  |-  ( ph  ->  { x  e.  A  |  B  e.  ( _V  \  { Z }
) }  =  {
x  e.  A  |  B  =/=  Z } )
124, 11eqtrd 2495 1  |-  ( ph  ->  ( F supp  Z )  =  { x  e.  A  |  B  =/= 
Z } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803   _Vcvv 3078    \ cdif 3434   {csn 3986    |-> cmpt 4459  (class class class)co 6201   supp csupp 6801
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-supp 6802
This theorem is referenced by:  rmsupp0  30930  domnmsuppn0  30931  rmsuppss  30932  suppmptcfin  30942  lcoc0  31089  linc1  31092
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