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Theorem suppimacnvss 6935
Description: The support of functions "defined" by inverse images is a subset of the support defined by df-supp 6926. (Contributed by AV, 7-Apr-2019.)
Assertion
Ref Expression
suppimacnvss  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( `' R "
( _V  \  { Z } ) )  C_  ( R supp  Z )
)

Proof of Theorem suppimacnvss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exsimpl 1723 . . . . 5  |-  ( E. y ( x R y  /\  y  =/= 
Z )  ->  E. y  x R y )
2 pm5.1 865 . . . . . 6  |-  ( ( x R y  /\  y  =/=  Z )  -> 
( x R y  <-> 
y  =/=  Z ) )
32eximi 1701 . . . . 5  |-  ( E. y ( x R y  /\  y  =/= 
Z )  ->  E. y
( x R y  <-> 
y  =/=  Z ) )
41, 3jca 534 . . . 4  |-  ( E. y ( x R y  /\  y  =/= 
Z )  ->  ( E. y  x R
y  /\  E. y
( x R y  <-> 
y  =/=  Z ) ) )
54a1i 11 . . 3  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( E. y ( x R y  /\  y  =/=  Z )  -> 
( E. y  x R y  /\  E. y ( x R y  <->  y  =/=  Z
) ) ) )
65ss2abdv 3534 . 2  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  { x  |  E. y ( x R y  /\  y  =/= 
Z ) }  C_  { x  |  ( E. y  x R y  /\  E. y ( x R y  <->  y  =/=  Z ) ) } )
7 cnvimadfsn 6934 . . 3  |-  ( `' R " ( _V 
\  { Z }
) )  =  {
x  |  E. y
( x R y  /\  y  =/=  Z
) }
87a1i 11 . 2  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( `' R "
( _V  \  { Z } ) )  =  { x  |  E. y ( x R y  /\  y  =/= 
Z ) } )
9 suppvalbr 6929 . 2  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R supp  Z )  =  { x  |  ( E. y  x R y  /\  E. y ( x R y  <->  y  =/=  Z
) ) } )
106, 8, 93sstr4d 3507 1  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( `' R "
( _V  \  { Z } ) )  C_  ( R supp  Z )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2407    =/= wne 2614   _Vcvv 3080    \ cdif 3433    C_ wss 3436   {csn 3998   class class class wbr 4423   `'ccnv 4852   "cima 4856  (class class class)co 6305   supp csupp 6925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-supp 6926
This theorem is referenced by:  suppimacnv  6936
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