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Theorem suppssov1 6091
Description: Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
suppssov1.s  |-  ( ph  ->  ( `' ( x  e.  D  |->  A )
" ( _V  \  { Y } ) ) 
C_  L )
suppssov1.o  |-  ( (
ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )
suppssov1.a  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  V )
suppssov1.b  |-  ( (
ph  /\  x  e.  D )  ->  B  e.  R )
Assertion
Ref Expression
suppssov1  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( A O B ) )
" ( _V  \  { Z } ) ) 
C_  L )
Distinct variable groups:    ph, v    ph, x    v, B    v, O    v, R    v, Y    x, Y    v, Z    x, Z
Allowed substitution hints:    A( x, v)    B( x)    D( x, v)    R( x)    L( x, v)    O( x)    V( x, v)

Proof of Theorem suppssov1
StepHypRef Expression
1 suppssov1.a . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  V )
2 elex 2809 . . . . . . . 8  |-  ( A  e.  V  ->  A  e.  _V )
31, 2syl 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  _V )
43adantr 451 . . . . . 6  |-  ( ( ( ph  /\  x  e.  D )  /\  ( A O B )  e.  ( _V  \  { Z } ) )  ->  A  e.  _V )
5 eldifsni 3763 . . . . . . . 8  |-  ( ( A O B )  e.  ( _V  \  { Z } )  -> 
( A O B )  =/=  Z )
6 suppssov1.b . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  D )  ->  B  e.  R )
7 suppssov1.o . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )
87ralrimiva 2639 . . . . . . . . . . . 12  |-  ( ph  ->  A. v  e.  R  ( Y O v )  =  Z )
98adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  D )  ->  A. v  e.  R  ( Y O v )  =  Z )
10 oveq2 5882 . . . . . . . . . . . . 13  |-  ( v  =  B  ->  ( Y O v )  =  ( Y O B ) )
1110eqeq1d 2304 . . . . . . . . . . . 12  |-  ( v  =  B  ->  (
( Y O v )  =  Z  <->  ( Y O B )  =  Z ) )
1211rspcva 2895 . . . . . . . . . . 11  |-  ( ( B  e.  R  /\  A. v  e.  R  ( Y O v )  =  Z )  -> 
( Y O B )  =  Z )
136, 9, 12syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  D )  ->  ( Y O B )  =  Z )
14 oveq1 5881 . . . . . . . . . . 11  |-  ( A  =  Y  ->  ( A O B )  =  ( Y O B ) )
1514eqeq1d 2304 . . . . . . . . . 10  |-  ( A  =  Y  ->  (
( A O B )  =  Z  <->  ( Y O B )  =  Z ) )
1613, 15syl5ibrcom 213 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  D )  ->  ( A  =  Y  ->  ( A O B )  =  Z ) )
1716necon3d 2497 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  (
( A O B )  =/=  Z  ->  A  =/=  Y ) )
185, 17syl5 28 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( A O B )  e.  ( _V 
\  { Z }
)  ->  A  =/=  Y ) )
1918imp 418 . . . . . 6  |-  ( ( ( ph  /\  x  e.  D )  /\  ( A O B )  e.  ( _V  \  { Z } ) )  ->  A  =/=  Y )
20 eldifsn 3762 . . . . . 6  |-  ( A  e.  ( _V  \  { Y } )  <->  ( A  e.  _V  /\  A  =/= 
Y ) )
214, 19, 20sylanbrc 645 . . . . 5  |-  ( ( ( ph  /\  x  e.  D )  /\  ( A O B )  e.  ( _V  \  { Z } ) )  ->  A  e.  ( _V  \  { Y } ) )
2221ex 423 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( A O B )  e.  ( _V 
\  { Z }
)  ->  A  e.  ( _V  \  { Y } ) ) )
2322ss2rabdv 3267 . . 3  |-  ( ph  ->  { x  e.  D  |  ( A O B )  e.  ( _V  \  { Z } ) }  C_  { x  e.  D  |  A  e.  ( _V  \  { Y } ) } )
24 eqid 2296 . . . 4  |-  ( x  e.  D  |->  ( A O B ) )  =  ( x  e.  D  |->  ( A O B ) )
2524mptpreima 5182 . . 3  |-  ( `' ( x  e.  D  |->  ( A O B ) ) " ( _V  \  { Z }
) )  =  {
x  e.  D  | 
( A O B )  e.  ( _V 
\  { Z }
) }
26 eqid 2296 . . . 4  |-  ( x  e.  D  |->  A )  =  ( x  e.  D  |->  A )
2726mptpreima 5182 . . 3  |-  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
) )  =  {
x  e.  D  |  A  e.  ( _V  \  { Y } ) }
2823, 25, 273sstr4g 3232 . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( A O B ) )
" ( _V  \  { Z } ) ) 
C_  ( `' ( x  e.  D  |->  A ) " ( _V 
\  { Y }
) ) )
29 suppssov1.s . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  A )
" ( _V  \  { Y } ) ) 
C_  L )
3028, 29sstrd 3202 1  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( A O B ) )
" ( _V  \  { Z } ) ) 
C_  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653    e. cmpt 4093   `'ccnv 4704   "cima 4708  (class class class)co 5874
This theorem is referenced by:  suppssof1  6135  ply1coe  16384  evlslem6  19413  plypf1  19610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877
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