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Theorem suppssof1 6103
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  oF 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 5367 . . . . . 6  |-  ( A : D --> V  ->  A  Fn  D )
31, 2syl 14 . . . . 5  |-  ( ph  ->  A  Fn  D )
4 suppssof1.b . . . . . 6  |-  ( ph  ->  B : D --> R )
5 ffn 5367 . . . . . 6  |-  ( B : D --> R  ->  B  Fn  D )
64, 5syl 14 . . . . 5  |-  ( ph  ->  B  Fn  D )
7 suppssof1.d . . . . 5  |-  ( ph  ->  D  e.  W )
8 inidm 3346 . . . . 5  |-  ( D  i^i  D )  =  D
9 eqidd 2178 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  =  ( A `  x ) )
10 eqidd 2178 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  =  ( B `  x ) )
113, 6, 7, 7, 8, 9, 10offval 6093 . . . 4  |-  ( ph  ->  ( A  oF O B )  =  ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) )
1211cnveqd 4805 . . 3  |-  ( ph  ->  `' ( A  oF O B )  =  `' ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) )
1312imaeq1d 4971 . 2  |-  ( ph  ->  ( `' ( A  oF O B ) " ( _V 
\  { Z }
) )  =  ( `' ( x  e.  D  |->  ( ( A `
 x ) O ( B `  x
) ) ) "
( _V  \  { Z } ) ) )
141feqmptd 5572 . . . . . 6  |-  ( ph  ->  A  =  ( x  e.  D  |->  ( A `
 x ) ) )
1514cnveqd 4805 . . . . 5  |-  ( ph  ->  `' A  =  `' ( x  e.  D  |->  ( A `  x
) ) )
1615imaeq1d 4971 . . . 4  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  =  ( `' ( x  e.  D  |->  ( A `
 x ) )
" ( _V  \  { Y } ) ) )
17 suppssof1.s . . . 4  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  C_  L )
1816, 17eqsstrrd 3194 . . 3  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( A `
 x ) )
" ( _V  \  { Y } ) ) 
C_  L )
19 suppssof1.o . . 3  |-  ( (
ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )
20 funfvex 5534 . . . . 5  |-  ( ( Fun  A  /\  x  e.  dom  A )  -> 
( A `  x
)  e.  _V )
2120funfni 5318 . . . 4  |-  ( ( A  Fn  D  /\  x  e.  D )  ->  ( A `  x
)  e.  _V )
223, 21sylan 283 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  e.  _V )
234ffvelcdmda 5654 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  e.  R )
2418, 19, 22, 23suppssov1 6083 . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) )
" ( _V  \  { Z } ) ) 
C_  L )
2513, 24eqsstrd 3193 1  |-  ( ph  ->  ( `' ( A  oF O B ) " ( _V 
\  { Z }
) )  C_  L
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2739    \ cdif 3128    C_ wss 3131   {csn 3594    |-> cmpt 4066   `'ccnv 4627   "cima 4631    Fn wfn 5213   -->wf 5214   ` cfv 5218  (class class class)co 5878    oFcof 6084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-of 6086
This theorem is referenced by: (None)
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