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Theorem suppssfv 6330
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 3952 . . . . 5  |-  ( ( F `  A )  e.  ( _V  \  { Z } )  -> 
( F `  A
)  =/=  Z )
2 suppssfv.v . . . . . . . . 9  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  V )
3 elex 2970 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  _V )
42, 3syl 16 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  _V )
54adantr 453 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  e.  _V )
6 suppssfv.f . . . . . . . . . . 11  |-  ( ph  ->  ( F `  Y
)  =  Z )
7 fveq2 5757 . . . . . . . . . . . 12  |-  ( A  =  Y  ->  ( F `  A )  =  ( F `  Y ) )
87eqeq1d 2450 . . . . . . . . . . 11  |-  ( A  =  Y  ->  (
( F `  A
)  =  Z  <->  ( F `  Y )  =  Z ) )
96, 8syl5ibrcom 215 . . . . . . . . . 10  |-  ( ph  ->  ( A  =  Y  ->  ( F `  A )  =  Z ) )
109necon3d 2645 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  A )  =/=  Z  ->  A  =/=  Y ) )
1110adantr 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  (
( F `  A
)  =/=  Z  ->  A  =/=  Y ) )
1211imp 420 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  =/=  Y )
13 eldifsn 3951 . . . . . . 7  |-  ( A  e.  ( _V  \  { Y } )  <->  ( A  e.  _V  /\  A  =/= 
Y ) )
145, 12, 13sylanbrc 647 . . . . . 6  |-  ( ( ( ph  /\  x  e.  D )  /\  ( F `  A )  =/=  Z )  ->  A  e.  ( _V  \  { Y } ) )
1514ex 425 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  (
( F `  A
)  =/=  Z  ->  A  e.  ( _V  \  { Y } ) ) )
161, 15syl5 31 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( F `  A
)  e.  ( _V 
\  { Z }
)  ->  A  e.  ( _V  \  { Y } ) ) )
1716ss2rabdv 3410 . . 3  |-  ( ph  ->  { x  e.  D  |  ( F `  A )  e.  ( _V  \  { Z } ) }  C_  { x  e.  D  |  A  e.  ( _V  \  { Y } ) } )
18 eqid 2442 . . . 4  |-  ( x  e.  D  |->  ( F `
 A ) )  =  ( x  e.  D  |->  ( F `  A ) )
1918mptpreima 5392 . . 3  |-  ( `' ( x  e.  D  |->  ( F `  A
) ) " ( _V  \  { Z }
) )  =  {
x  e.  D  | 
( F `  A
)  e.  ( _V 
\  { Z }
) }
20 eqid 2442 . . . 4  |-  ( x  e.  D  |->  A )  =  ( x  e.  D  |->  A )
2120mptpreima 5392 . . 3  |-  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
) )  =  {
x  e.  D  |  A  e.  ( _V  \  { Y } ) }
2217, 19, 213sstr4g 3375 . 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 3344 1  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( F `
 A ) )
" ( _V  \  { Z } ) ) 
C_  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   {crab 2715   _Vcvv 2962    \ cdif 3303    C_ wss 3306   {csn 3838    e. cmpt 4291   `'ccnv 4906   "cima 4910   ` cfv 5483
This theorem is referenced by:  evlslem2  16599  evlslem6  19965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-xp 4913  df-rel 4914  df-cnv 4915  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fv 5491
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