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Theorem suppssof1 6135
Description: Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
suppssof1.s  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  C_  L )
suppssof1.o  |-  ( (
ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )
suppssof1.a  |-  ( ph  ->  A : D --> V )
suppssof1.b  |-  ( ph  ->  B : D --> R )
suppssof1.d  |-  ( ph  ->  D  e.  W )
Assertion
Ref Expression
suppssof1  |-  ( ph  ->  ( `' ( A  o F O B ) " ( _V 
\  { Z }
) )  C_  L
)
Distinct variable groups:    ph, v    v, B    v, O    v, R    v, Y    v, Z
Allowed substitution hints:    A( v)    D( v)    L( v)    V( v)    W( v)

Proof of Theorem suppssof1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 suppssof1.a . . . . . 6  |-  ( ph  ->  A : D --> V )
2 ffn 5405 . . . . . 6  |-  ( A : D --> V  ->  A  Fn  D )
31, 2syl 15 . . . . 5  |-  ( ph  ->  A  Fn  D )
4 suppssof1.b . . . . . 6  |-  ( ph  ->  B : D --> R )
5 ffn 5405 . . . . . 6  |-  ( B : D --> R  ->  B  Fn  D )
64, 5syl 15 . . . . 5  |-  ( ph  ->  B  Fn  D )
7 suppssof1.d . . . . 5  |-  ( ph  ->  D  e.  W )
8 inidm 3391 . . . . 5  |-  ( D  i^i  D )  =  D
9 eqidd 2297 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  =  ( A `  x ) )
10 eqidd 2297 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  =  ( B `  x ) )
113, 6, 7, 7, 8, 9, 10offval 6101 . . . 4  |-  ( ph  ->  ( A  o F O B )  =  ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) )
1211cnveqd 4873 . . 3  |-  ( ph  ->  `' ( A  o F O B )  =  `' ( x  e.  D  |->  ( ( A `
 x ) O ( B `  x
) ) ) )
1312imaeq1d 5027 . 2  |-  ( ph  ->  ( `' ( A  o F O B ) " ( _V 
\  { Z }
) )  =  ( `' ( x  e.  D  |->  ( ( A `
 x ) O ( B `  x
) ) ) "
( _V  \  { Z } ) ) )
141feqmptd 5591 . . . . . 6  |-  ( ph  ->  A  =  ( x  e.  D  |->  ( A `
 x ) ) )
1514cnveqd 4873 . . . . 5  |-  ( ph  ->  `' A  =  `' ( x  e.  D  |->  ( A `  x
) ) )
1615imaeq1d 5027 . . . 4  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  =  ( `' ( x  e.  D  |->  ( A `
 x ) )
" ( _V  \  { Y } ) ) )
17 suppssof1.s . . . 4  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  C_  L )
1816, 17eqsstr3d 3226 . . 3  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( A `
 x ) )
" ( _V  \  { Y } ) ) 
C_  L )
19 suppssof1.o . . 3  |-  ( (
ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )
20 fvex 5555 . . . 4  |-  ( A `
 x )  e. 
_V
2120a1i 10 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  e.  _V )
22 ffvelrn 5679 . . . 4  |-  ( ( B : D --> R  /\  x  e.  D )  ->  ( B `  x
)  e.  R )
234, 22sylan 457 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  e.  R )
2418, 19, 21, 23suppssov1 6091 . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) )
" ( _V  \  { Z } ) ) 
C_  L )
2513, 24eqsstrd 3225 1  |-  ( ph  ->  ( `' ( A  o F O B ) " ( _V 
\  { Z }
) )  C_  L
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653    e. cmpt 4093   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  psrbagev1  16263  jensen  20299
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-rep 4147  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094
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