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Theorem suppssfv 6260
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 3888 . . . . 5  |-  ( ( F `  A )  e.  ( _V  \  { Z } )  -> 
( F `  A
)  =/=  Z )
2 suppssfv.v . . . . . . . . 9  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  V )
3 elex 2924 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  _V )
42, 3syl 16 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  _V )
54adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  e.  _V )
6 suppssfv.f . . . . . . . . . . 11  |-  ( ph  ->  ( F `  Y
)  =  Z )
7 fveq2 5687 . . . . . . . . . . . 12  |-  ( A  =  Y  ->  ( F `  A )  =  ( F `  Y ) )
87eqeq1d 2412 . . . . . . . . . . 11  |-  ( A  =  Y  ->  (
( F `  A
)  =  Z  <->  ( F `  Y )  =  Z ) )
96, 8syl5ibrcom 214 . . . . . . . . . 10  |-  ( ph  ->  ( A  =  Y  ->  ( F `  A )  =  Z ) )
109necon3d 2605 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  A )  =/=  Z  ->  A  =/=  Y ) )
1110adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  (
( F `  A
)  =/=  Z  ->  A  =/=  Y ) )
1211imp 419 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  =/=  Y )
13 eldifsn 3887 . . . . . . 7  |-  ( A  e.  ( _V  \  { Y } )  <->  ( A  e.  _V  /\  A  =/= 
Y ) )
145, 12, 13sylanbrc 646 . . . . . 6  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  e.  ( _V  \  { Y } ) )
1514ex 424 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  (
( F `  A
)  =/=  Z  ->  A  e.  ( _V  \  { Y } ) ) )
161, 15syl5 30 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( F `  A
)  e.  ( _V 
\  { Z }
)  ->  A  e.  ( _V  \  { Y } ) ) )
1716ss2rabdv 3384 . . 3  |-  ( ph  ->  { x  e.  D  |  ( F `  A )  e.  ( _V  \  { Z } ) }  C_  { x  e.  D  |  A  e.  ( _V  \  { Y } ) } )
18 eqid 2404 . . . 4  |-  ( x  e.  D  |->  ( F `
 A ) )  =  ( x  e.  D  |->  ( F `  A ) )
1918mptpreima 5322 . . 3  |-  ( `' ( x  e.  D  |->  ( F `  A
) ) " ( _V  \  { Z }
) )  =  {
x  e.  D  | 
( F `  A
)  e.  ( _V 
\  { Z }
) }
20 eqid 2404 . . . 4  |-  ( x  e.  D  |->  A )  =  ( x  e.  D  |->  A )
2120mptpreima 5322 . . 3  |-  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
) )  =  {
x  e.  D  |  A  e.  ( _V  \  { Y } ) }
2217, 19, 213sstr4g 3349 . 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 3318 1  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( F `
 A ) )
" ( _V  \  { Z } ) ) 
C_  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774    e. cmpt 4226   `'ccnv 4836   "cima 4840   ` cfv 5413
This theorem is referenced by:  evlslem2  16523  evlslem6  19887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421
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