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Theorem suppssov1 6075
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 2796 . . . . . . . 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 3750 . . . . . . . 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 2626 . . . . . . . . . . . 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 5866 . . . . . . . . . . . . 13  |-  ( v  =  B  ->  ( Y O v )  =  ( Y O B ) )
1110eqeq1d 2291 . . . . . . . . . . . 12  |-  ( v  =  B  ->  (
( Y O v )  =  Z  <->  ( Y O B )  =  Z ) )
1211rspcva 2882 . . . . . . . . . . 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 5865 . . . . . . . . . . 11  |-  ( A  =  Y  ->  ( A O B )  =  ( Y O B ) )
1514eqeq1d 2291 . . . . . . . . . 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 2484 . . . . . . . 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 3749 . . . . . 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 3254 . . 3  |-  ( ph  ->  { x  e.  D  |  ( A O B )  e.  ( _V  \  { Z } ) }  C_  { x  e.  D  |  A  e.  ( _V  \  { Y } ) } )
24 eqid 2283 . . . 4  |-  ( x  e.  D  |->  ( A O B ) )  =  ( x  e.  D  |->  ( A O B ) )
2524mptpreima 5166 . . 3  |-  ( `' ( x  e.  D  |->  ( A O B ) ) " ( _V  \  { Z }
) )  =  {
x  e.  D  | 
( A O B )  e.  ( _V 
\  { Z }
) }
26 eqid 2283 . . . 4  |-  ( x  e.  D  |->  A )  =  ( x  e.  D  |->  A )
2726mptpreima 5166 . . 3  |-  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
) )  =  {
x  e.  D  |  A  e.  ( _V  \  { Y } ) }
2823, 25, 273sstr4g 3219 . 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 3189 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692  (class class class)co 5858
This theorem is referenced by:  suppssof1  6119  ply1coe  16368  evlslem6  19397  plypf1  19594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861
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