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Theorem suppssof1 6119
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 5389 . . . . . 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 5389 . . . . . 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 3378 . . . . 5  |-  ( D  i^i  D )  =  D
9 eqidd 2284 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  =  ( A `  x ) )
10 eqidd 2284 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  =  ( B `  x ) )
113, 6, 7, 7, 8, 9, 10offval 6085 . . . 4  |-  ( ph  ->  ( A  o F O B )  =  ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) )
1211cnveqd 4857 . . 3  |-  ( ph  ->  `' ( A  o F O B )  =  `' ( x  e.  D  |->  ( ( A `
 x ) O ( B `  x
) ) ) )
1312imaeq1d 5011 . 2  |-  ( ph  ->  ( `' ( A  o F O B ) " ( _V 
\  { Z }
) )  =  ( `' ( x  e.  D  |->  ( ( A `
 x ) O ( B `  x
) ) ) "
( _V  \  { Z } ) ) )
141feqmptd 5575 . . . . . 6  |-  ( ph  ->  A  =  ( x  e.  D  |->  ( A `
 x ) ) )
1514cnveqd 4857 . . . . 5  |-  ( ph  ->  `' A  =  `' ( x  e.  D  |->  ( A `  x
) ) )
1615imaeq1d 5011 . . . 4  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  =  ( `' ( x  e.  D  |->  ( A `
 x ) )
" ( _V  \  { Y } ) ) )
17 suppssof1.s . . . 4  |-  ( ph  ->  ( `' A "
( _V  \  { Y } ) )  C_  L )
1816, 17eqsstr3d 3213 . . 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 5539 . . . 4  |-  ( A `
 x )  e. 
_V
2120a1i 10 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  e.  _V )
22 ffvelrn 5663 . . . 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 6075 . 2  |-  ( ph  ->  ( `' ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) )
" ( _V  \  { Z } ) ) 
C_  L )
2513, 24eqsstrd 3212 1  |-  ( ph  ->  ( `' ( A  o F O B ) " ( _V 
\  { Z }
) )  C_  L
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  psrbagev1  16247  jensen  20283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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