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Theorem suppssov1 5947
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 2671 . . . . . . . 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 3622 . . . . . . . 8  |-  ( ( A O B )  e.  ( _V  \  { Z } )  -> 
( A O B )  =/=  Z )
6 oveq2 5750 . . . . . . . . . . . 12  |-  ( v  =  B  ->  ( Y O v )  =  ( Y O B ) )
76eqeq1d 2126 . . . . . . . . . . 11  |-  ( v  =  B  ->  (
( Y O v )  =  Z  <->  ( Y O B )  =  Z ) )
8 suppssov1.o . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )
98ralrimiva 2482 . . . . . . . . . . . 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 2768 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  D )  ->  ( Y O B )  =  Z )
13 oveq1 5749 . . . . . . . . . . 11  |-  ( A  =  Y  ->  ( A O B )  =  ( Y O B ) )
1413eqeq1d 2126 . . . . . . . . . 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 2329 . . . . . . . 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 3620 . . . . . 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 3148 . . 3  |-  ( ph  ->  { x  e.  D  |  ( A O B )  e.  ( _V  \  { Z } ) }  C_  { x  e.  D  |  A  e.  ( _V  \  { Y } ) } )
23 eqid 2117 . . . 4  |-  ( x  e.  D  |->  ( A O B ) )  =  ( x  e.  D  |->  ( A O B ) )
2423mptpreima 5002 . . 3  |-  ( `' ( x  e.  D  |->  ( A O B ) ) " ( _V  \  { Z }
) )  =  {
x  e.  D  | 
( A O B )  e.  ( _V 
\  { Z }
) }
25 eqid 2117 . . . 4  |-  ( x  e.  D  |->  A )  =  ( x  e.  D  |->  A )
2625mptpreima 5002 . . 3  |-  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
) )  =  {
x  e.  D  |  A  e.  ( _V  \  { Y } ) }
2722, 24, 263sstr4g 3110 . 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 3077 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 1316    e. wcel 1465    =/= wne 2285   A.wral 2393   {crab 2397   _Vcvv 2660    \ cdif 3038    C_ wss 3041   {csn 3497    |-> cmpt 3959   `'ccnv 4508   "cima 4512  (class class class)co 5742
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  suppssof1  5967
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