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Theorem ressval2 12029
Description: Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypotheses
Ref Expression
ressbas.r  |-  R  =  ( Ws  A )
ressbas.b  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
ressval2  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  R  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. ) )

Proof of Theorem ressval2
Dummy variables  w  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ressbas.r . 2  |-  R  =  ( Ws  A )
2 simp2 982 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  W  e.  X )
32elexd 2699 . . . 4  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  W  e.  _V )
4 simp3 983 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  A  e.  Y )
54elexd 2699 . . . 4  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  A  e.  _V )
6 simp1 981 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  -.  B  C_  A )
76iffalsed 3484 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  if ( B  C_  A ,  W ,  ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. ) )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B
) >. ) )
8 basendxnn 12024 . . . . . . 7  |-  ( Base `  ndx )  e.  NN
98a1i 9 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  ( Base ` 
ndx )  e.  NN )
10 inex1g 4064 . . . . . . 7  |-  ( A  e.  Y  ->  ( A  i^i  B )  e. 
_V )
114, 10syl 14 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  ( A  i^i  B )  e.  _V )
12 setsex 12001 . . . . . 6  |-  ( ( W  e.  X  /\  ( Base `  ndx )  e.  NN  /\  ( A  i^i  B )  e. 
_V )  ->  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )  e.  _V )
132, 9, 11, 12syl3anc 1216 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. )  e.  _V )
147, 13eqeltrd 2216 . . . 4  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  if ( B  C_  A ,  W ,  ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. ) )  e. 
_V )
15 simpl 108 . . . . . . . . 9  |-  ( ( w  =  W  /\  a  =  A )  ->  w  =  W )
1615fveq2d 5425 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  ( Base `  W ) )
17 ressbas.b . . . . . . . 8  |-  B  =  ( Base `  W
)
1816, 17syl6eqr 2190 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  B )
19 simpr 109 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  a  =  A )
2018, 19sseq12d 3128 . . . . . 6  |-  ( ( w  =  W  /\  a  =  A )  ->  ( ( Base `  w
)  C_  a  <->  B  C_  A
) )
2119, 18ineq12d 3278 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( a  i^i  ( Base `  w ) )  =  ( A  i^i  B ) )
2221opeq2d 3712 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  -> 
<. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w
) ) >.  =  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. )
2315, 22oveq12d 5792 . . . . . 6  |-  ( ( w  =  W  /\  a  =  A )  ->  ( w sSet  <. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w ) )
>. )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
2420, 15, 23ifbieq12d 3498 . . . . 5  |-  ( ( w  =  W  /\  a  =  A )  ->  if ( ( Base `  w )  C_  a ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w
) ) >. )
)  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
25 df-ress 11977 . . . . 5  |-s  =  ( w  e.  _V ,  a  e. 
_V  |->  if ( (
Base `  w )  C_  a ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w ) )
>. ) ) )
2624, 25ovmpoga 5900 . . . 4  |-  ( ( W  e.  _V  /\  A  e.  _V  /\  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) )  e. 
_V )  ->  ( Ws  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
) )
273, 5, 14, 26syl3anc 1216 . . 3  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  ( Ws  A
)  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
2827, 7eqtrd 2172 . 2  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  ( Ws  A
)  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
291, 28syl5eq 2184 1  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  R  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   _Vcvv 2686    i^i cin 3070    C_ wss 3071   ifcif 3474   <.cop 3530   ` cfv 5123  (class class class)co 5774   NNcn 8727   ndxcnx 11966   sSet csts 11967   Basecbs 11969   ↾s cress 11970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1re 7721  ax-addrcl 7724
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-inn 8728  df-ndx 11972  df-slot 11973  df-base 11975  df-sets 11976  df-ress 11977
This theorem is referenced by: (None)
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