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Theorem suppssfv 5946
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 3622 . . . . 5  |-  ( ( F `  A )  e.  ( _V  \  { Z } )  -> 
( F `  A
)  =/=  Z )
2 suppssfv.v . . . . . . . . 9  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  V )
3 elex 2671 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  _V )
42, 3syl 14 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  _V )
54adantr 274 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  e.  _V )
6 suppssfv.f . . . . . . . . . . 11  |-  ( ph  ->  ( F `  Y
)  =  Z )
7 fveq2 5389 . . . . . . . . . . . 12  |-  ( A  =  Y  ->  ( F `  A )  =  ( F `  Y ) )
87eqeq1d 2126 . . . . . . . . . . 11  |-  ( A  =  Y  ->  (
( F `  A
)  =  Z  <->  ( F `  Y )  =  Z ) )
96, 8syl5ibrcom 156 . . . . . . . . . 10  |-  ( ph  ->  ( A  =  Y  ->  ( F `  A )  =  Z ) )
109necon3d 2329 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  A )  =/=  Z  ->  A  =/=  Y ) )
1110adantr 274 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  (
( F `  A
)  =/=  Z  ->  A  =/=  Y ) )
1211imp 123 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  =/=  Y )
13 eldifsn 3620 . . . . . . 7  |-  ( A  e.  ( _V  \  { Y } )  <->  ( A  e.  _V  /\  A  =/= 
Y ) )
145, 12, 13sylanbrc 413 . . . . . 6  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  e.  ( _V  \  { Y } ) )
1514ex 114 . . . . 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 3148 . . 3  |-  ( ph  ->  { x  e.  D  |  ( F `  A )  e.  ( _V  \  { Z } ) }  C_  { x  e.  D  |  A  e.  ( _V  \  { Y } ) } )
18 eqid 2117 . . . 4  |-  ( x  e.  D  |->  ( F `
 A ) )  =  ( x  e.  D  |->  ( F `  A ) )
1918mptpreima 5002 . . 3  |-  ( `' ( x  e.  D  |->  ( F `  A
) ) " ( _V  \  { Z }
) )  =  {
x  e.  D  | 
( F `  A
)  e.  ( _V 
\  { Z }
) }
20 eqid 2117 . . . 4  |-  ( x  e.  D  |->  A )  =  ( x  e.  D  |->  A )
2120mptpreima 5002 . . 3  |-  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
) )  =  {
x  e.  D  |  A  e.  ( _V  \  { Y } ) }
2217, 19, 213sstr4g 3110 . 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 3077 1  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( F `
 A ) )
" ( _V  \  { Z } ) ) 
C_  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465    =/= wne 2285   {crab 2397   _Vcvv 2660    \ cdif 3038    C_ wss 3041   {csn 3497    |-> cmpt 3959   `'ccnv 4508   "cima 4512   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fv 5101
This theorem is referenced by: (None)
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