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Theorem mapdhval0 32525
Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
Hypotheses
Ref Expression
mapdh.q  |-  Q  =  ( 0g `  C
)
mapdh.i  |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
mapdh0.o  |-  .0.  =  ( 0g `  U )
mapdh0.x  |-  ( ph  ->  X  e.  A )
mapdh0.f  |-  ( ph  ->  F  e.  B )
Assertion
Ref Expression
mapdhval0  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
Distinct variable groups:    x, D    x, h, F    x, J    x, M    x, N    x,  .0.    x, Q    x, R    x, 
.-    h, X, x    ph, h    .0. ,
h
Allowed substitution hints:    ph( x)    A( x, h)    B( x, h)    C( x, h)    D( h)    Q( h)    R( h)    U( x, h)    I( x, h)    J( h)    M( h)    .- ( h)    N( h)

Proof of Theorem mapdhval0
StepHypRef Expression
1 mapdh.q . . 3  |-  Q  =  ( 0g `  C
)
2 mapdh.i . . 3  |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
3 mapdh0.x . . 3  |-  ( ph  ->  X  e.  A )
4 mapdh0.f . . 3  |-  ( ph  ->  F  e.  B )
5 mapdh0.o . . . . 5  |-  .0.  =  ( 0g `  U )
6 fvex 5744 . . . . 5  |-  ( 0g
`  U )  e. 
_V
75, 6eqeltri 2508 . . . 4  |-  .0.  e.  _V
87a1i 11 . . 3  |-  ( ph  ->  .0.  e.  _V )
91, 2, 3, 4, 8mapdhval 32524 . 2  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  if (  .0.  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 {  .0.  }
) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  .0.  ) } ) )  =  ( J `  { ( F R h ) } ) ) ) ) )
10 eqid 2438 . . 3  |-  .0.  =  .0.
11 iftrue 3747 . . 3  |-  (  .0.  =  .0.  ->  if (  .0.  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 {  .0.  }
) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  .0.  ) } ) )  =  ( J `  { ( F R h ) } ) ) ) )  =  Q )
1210, 11ax-mp 8 . 2  |-  if (  .0.  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 {  .0.  }
) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  .0.  ) } ) )  =  ( J `  { ( F R h ) } ) ) ) )  =  Q
139, 12syl6eq 2486 1  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   ifcif 3741   {csn 3816   <.cotp 3820    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   iota_crio 6544   0gc0g 13725
This theorem is referenced by:  mapdhcl  32527  mapdh6bN  32537  mapdh6cN  32538  mapdh6dN  32539  mapdh8  32589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-ot 3826  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-1st 6351  df-2nd 6352  df-riota 6551
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