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Theorem suppssov1 5972
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 2692 . . . . . . . 8  |-  ( A  e.  V  ->  A  e.  _V )
31, 2syl 14 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  A  e.  _V )
43adantr 274 . . . . . 6  |-  ( ( ( ph  /\  x  e.  D )  /\  ( A O B )  e.  ( _V  \  { Z } ) )  ->  A  e.  _V )
5 eldifsni 3647 . . . . . . . 8  |-  ( ( A O B )  e.  ( _V  \  { Z } )  -> 
( A O B )  =/=  Z )
6 oveq2 5775 . . . . . . . . . . . 12  |-  ( v  =  B  ->  ( Y O v )  =  ( Y O B ) )
76eqeq1d 2146 . . . . . . . . . . 11  |-  ( v  =  B  ->  (
( Y O v )  =  Z  <->  ( Y O B )  =  Z ) )
8 suppssov1.o . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )
98ralrimiva 2503 . . . . . . . . . . . 12  |-  ( ph  ->  A. v  e.  R  ( Y O v )  =  Z )
109adantr 274 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  D )  ->  A. v  e.  R  ( Y O v )  =  Z )
11 suppssov1.b . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  D )  ->  B  e.  R )
127, 10, 11rspcdva 2789 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  D )  ->  ( Y O B )  =  Z )
13 oveq1 5774 . . . . . . . . . . 11  |-  ( A  =  Y  ->  ( A O B )  =  ( Y O B ) )
1413eqeq1d 2146 . . . . . . . . . 10  |-  ( A  =  Y  ->  (
( A O B )  =  Z  <->  ( Y O B )  =  Z ) )
1512, 14syl5ibrcom 156 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  D )  ->  ( A  =  Y  ->  ( A O B )  =  Z ) )
1615necon3d 2350 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  (
( A O B )  =/=  Z  ->  A  =/=  Y ) )
175, 16syl5 32 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( A O B )  e.  ( _V 
\  { Z }
)  ->  A  =/=  Y ) )
1817imp 123 . . . . . 6  |-  ( ( ( ph  /\  x  e.  D )  /\  ( A O B )  e.  ( _V  \  { Z } ) )  ->  A  =/=  Y )
19 eldifsn 3645 . . . . . 6  |-  ( A  e.  ( _V  \  { Y } )  <->  ( A  e.  _V  /\  A  =/= 
Y ) )
204, 18, 19sylanbrc 413 . . . . 5  |-  ( ( ( ph  /\  x  e.  D )  /\  ( A O B )  e.  ( _V  \  { Z } ) )  ->  A  e.  ( _V  \  { Y } ) )
2120ex 114 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( A O B )  e.  ( _V 
\  { Z }
)  ->  A  e.  ( _V  \  { Y } ) ) )
2221ss2rabdv 3173 . . 3  |-  ( ph  ->  { x  e.  D  |  ( A O B )  e.  ( _V  \  { Z } ) }  C_  { x  e.  D  |  A  e.  ( _V  \  { Y } ) } )
23 eqid 2137 . . . 4  |-  ( x  e.  D  |->  ( A O B ) )  =  ( x  e.  D  |->  ( A O B ) )
2423mptpreima 5027 . . 3  |-  ( `' ( x  e.  D  |->  ( A O B ) ) " ( _V  \  { Z }
) )  =  {
x  e.  D  | 
( A O B )  e.  ( _V 
\  { Z }
) }
25 eqid 2137 . . . 4  |-  ( x  e.  D  |->  A )  =  ( x  e.  D  |->  A )
2625mptpreima 5027 . . 3  |-  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
) )  =  {
x  e.  D  |  A  e.  ( _V  \  { Y } ) }
2722, 24, 263sstr4g 3135 . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( A O B ) )
" ( _V  \  { Z } ) ) 
C_  ( `' ( x  e.  D  |->  A ) " ( _V 
\  { Y }
) ) )
28 suppssov1.s . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  A )
" ( _V  \  { Y } ) ) 
C_  L )
2927, 28sstrd 3102 1  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( A O B ) )
" ( _V  \  { Z } ) ) 
C_  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    =/= wne 2306   A.wral 2414   {crab 2418   _Vcvv 2681    \ cdif 3063    C_ wss 3066   {csn 3522    |-> cmpt 3984   `'ccnv 4533   "cima 4537  (class class class)co 5767
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fv 5126  df-ov 5770
This theorem is referenced by:  suppssof1  5992
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