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Theorem suppssof1 5854
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 5147 . . . . . 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 5147 . . . . . 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 3207 . . . . 5  |-  ( D  i^i  D )  =  D
9 eqidd 2089 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  =  ( A `  x ) )
10 eqidd 2089 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  =  ( B `  x ) )
113, 6, 7, 7, 8, 9, 10offval 5845 . . . 4  |-  ( ph  ->  ( A  oF O B )  =  ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) )
1211cnveqd 4600 . . 3  |-  ( ph  ->  `' ( A  oF O B )  =  `' ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) )
1312imaeq1d 4760 . 2  |-  ( ph  ->  ( `' ( A  oF O B ) " ( _V 
\  { Z }
) )  =  ( `' ( x  e.  D  |->  ( ( A `
 x ) O ( B `  x
) ) ) "
( _V  \  { Z } ) ) )
141feqmptd 5341 . . . . . 6  |-  ( ph  ->  A  =  ( x  e.  D  |->  ( A `
 x ) ) )
1514cnveqd 4600 . . . . 5  |-  ( ph  ->  `' A  =  `' ( x  e.  D  |->  ( A `  x
) ) )
1615imaeq1d 4760 . . . 4  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  =  ( `' ( x  e.  D  |->  ( A `
 x ) )
" ( _V  \  { Y } ) ) )
17 suppssof1.s . . . 4  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  C_  L )
1816, 17eqsstr3d 3059 . . 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 5306 . . . . 5  |-  ( ( Fun  A  /\  x  e.  dom  A )  -> 
( A `  x
)  e.  _V )
2120funfni 5100 . . . 4  |-  ( ( A  Fn  D  /\  x  e.  D )  ->  ( A `  x
)  e.  _V )
223, 21sylan 277 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  e.  _V )
234ffvelrnda 5418 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  e.  R )
2418, 19, 22, 23suppssov1 5835 . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) )
" ( _V  \  { Z } ) ) 
C_  L )
2513, 24eqsstrd 3058 1  |-  ( ph  ->  ( `' ( A  oF O B ) " ( _V 
\  { Z }
) )  C_  L
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619    \ cdif 2994    C_ wss 2997   {csn 3441    |-> cmpt 3891   `'ccnv 4427   "cima 4431    Fn wfn 4997   -->wf 4998   ` cfv 5002  (class class class)co 5634    oFcof 5836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-of 5838
This theorem is referenced by: (None)
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