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Theorem suppssfv 5852
Description: Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
suppssfv.a  |-  ( ph  ->  ( `' ( x  e.  D  |->  A )
" ( _V  \  { Y } ) ) 
C_  L )
suppssfv.f  |-  ( ph  ->  ( F `  Y
)  =  Z )
suppssfv.v  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  V )
Assertion
Ref Expression
suppssfv  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( F `
 A ) )
" ( _V  \  { Z } ) ) 
C_  L )
Distinct variable groups:    ph, x    x, Y    x, Z
Allowed substitution hints:    A( x)    D( x)    F( x)    L( x)    V( x)

Proof of Theorem suppssfv
StepHypRef Expression
1 eldifsni 3569 . . . . 5  |-  ( ( F `  A )  e.  ( _V  \  { Z } )  -> 
( F `  A
)  =/=  Z )
2 suppssfv.v . . . . . . . . 9  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  V )
3 elex 2630 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  _V )
42, 3syl 14 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  _V )
54adantr 270 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  e.  _V )
6 suppssfv.f . . . . . . . . . . 11  |-  ( ph  ->  ( F `  Y
)  =  Z )
7 fveq2 5305 . . . . . . . . . . . 12  |-  ( A  =  Y  ->  ( F `  A )  =  ( F `  Y ) )
87eqeq1d 2096 . . . . . . . . . . 11  |-  ( A  =  Y  ->  (
( F `  A
)  =  Z  <->  ( F `  Y )  =  Z ) )
96, 8syl5ibrcom 155 . . . . . . . . . 10  |-  ( ph  ->  ( A  =  Y  ->  ( F `  A )  =  Z ) )
109necon3d 2299 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  A )  =/=  Z  ->  A  =/=  Y ) )
1110adantr 270 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  (
( F `  A
)  =/=  Z  ->  A  =/=  Y ) )
1211imp 122 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  =/=  Y )
13 eldifsn 3567 . . . . . . 7  |-  ( A  e.  ( _V  \  { Y } )  <->  ( A  e.  _V  /\  A  =/= 
Y ) )
145, 12, 13sylanbrc 408 . . . . . 6  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  e.  ( _V  \  { Y } ) )
1514ex 113 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  (
( F `  A
)  =/=  Z  ->  A  e.  ( _V  \  { Y } ) ) )
161, 15syl5 32 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( F `  A
)  e.  ( _V 
\  { Z }
)  ->  A  e.  ( _V  \  { Y } ) ) )
1716ss2rabdv 3102 . . 3  |-  ( ph  ->  { x  e.  D  |  ( F `  A )  e.  ( _V  \  { Z } ) }  C_  { x  e.  D  |  A  e.  ( _V  \  { Y } ) } )
18 eqid 2088 . . . 4  |-  ( x  e.  D  |->  ( F `
 A ) )  =  ( x  e.  D  |->  ( F `  A ) )
1918mptpreima 4924 . . 3  |-  ( `' ( x  e.  D  |->  ( F `  A
) ) " ( _V  \  { Z }
) )  =  {
x  e.  D  | 
( F `  A
)  e.  ( _V 
\  { Z }
) }
20 eqid 2088 . . . 4  |-  ( x  e.  D  |->  A )  =  ( x  e.  D  |->  A )
2120mptpreima 4924 . . 3  |-  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
) )  =  {
x  e.  D  |  A  e.  ( _V  \  { Y } ) }
2217, 19, 213sstr4g 3067 . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( F `
 A ) )
" ( _V  \  { Z } ) ) 
C_  ( `' ( x  e.  D  |->  A ) " ( _V 
\  { Y }
) ) )
23 suppssfv.a . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  A )
" ( _V  \  { Y } ) ) 
C_  L )
2422, 23sstrd 3035 1  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( F `
 A ) )
" ( _V  \  { Z } ) ) 
C_  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    =/= wne 2255   {crab 2363   _Vcvv 2619    \ cdif 2996    C_ wss 2999   {csn 3446    |-> cmpt 3899   `'ccnv 4437   "cima 4441   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fv 5023
This theorem is referenced by: (None)
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