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Theorem suppssof1 5780
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 5097 . . . . . 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 5097 . . . . . 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 3191 . . . . 5  |-  ( D  i^i  D )  =  D
9 eqidd 2084 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  =  ( A `  x ) )
10 eqidd 2084 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  =  ( B `  x ) )
113, 6, 7, 7, 8, 9, 10offval 5771 . . . 4  |-  ( ph  ->  ( A  oF O B )  =  ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) )
1211cnveqd 4559 . . 3  |-  ( ph  ->  `' ( A  oF O B )  =  `' ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) )
1312imaeq1d 4717 . 2  |-  ( ph  ->  ( `' ( A  oF O B ) " ( _V 
\  { Z }
) )  =  ( `' ( x  e.  D  |->  ( ( A `
 x ) O ( B `  x
) ) ) "
( _V  \  { Z } ) ) )
141feqmptd 5279 . . . . . 6  |-  ( ph  ->  A  =  ( x  e.  D  |->  ( A `
 x ) ) )
1514cnveqd 4559 . . . . 5  |-  ( ph  ->  `' A  =  `' ( x  e.  D  |->  ( A `  x
) ) )
1615imaeq1d 4717 . . . 4  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  =  ( `' ( x  e.  D  |->  ( A `
 x ) )
" ( _V  \  { Y } ) ) )
17 suppssof1.s . . . 4  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  C_  L )
1816, 17eqsstr3d 3043 . . 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 5244 . . . . 5  |-  ( ( Fun  A  /\  x  e.  dom  A )  -> 
( A `  x
)  e.  _V )
2120funfni 5050 . . . 4  |-  ( ( A  Fn  D  /\  x  e.  D )  ->  ( A `  x
)  e.  _V )
223, 21sylan 277 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  e.  _V )
234ffvelrnda 5355 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  e.  R )
2418, 19, 22, 23suppssov1 5761 . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) )
" ( _V  \  { Z } ) ) 
C_  L )
2513, 24eqsstrd 3042 1  |-  ( ph  ->  ( `' ( A  oF O B ) " ( _V 
\  { Z }
) )  C_  L
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2610    \ cdif 2979    C_ wss 2982   {csn 3416    |-> cmpt 3859   `'ccnv 4390   "cima 4394    Fn wfn 4947   -->wf 4948   ` cfv 4952  (class class class)co 5564    oFcof 5762
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-of 5764
This theorem is referenced by: (None)
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