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Theorem mapdhval0 32537
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 5555 . . . . 5  |-  ( 0g
`  U )  e. 
_V
75, 6eqeltri 2366 . . . 4  |-  .0.  e.  _V
87a1i 10 . . 3  |-  ( ph  ->  .0.  e.  _V )
91, 2, 3, 4, 8mapdhval 32536 . 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 2296 . . 3  |-  .0.  =  .0.
11 iftrue 3584 . . 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 2344 1  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   ifcif 3578   {csn 3653   <.cotp 3657    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   iota_crio 6313   0gc0g 13416
This theorem is referenced by:  mapdhcl  32539  mapdh6bN  32549  mapdh6cN  32550  mapdh6dN  32551  mapdh8  32601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320
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