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Theorem mapdhval0 31915
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 5539 . . . . 5  |-  ( 0g
`  U )  e. 
_V
75, 6eqeltri 2353 . . . 4  |-  .0.  e.  _V
87a1i 10 . . 3  |-  ( ph  ->  .0.  e.  _V )
91, 2, 3, 4, 8mapdhval 31914 . 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 2283 . . 3  |-  .0.  =  .0.
11 iftrue 3571 . . 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 2331 1  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   ifcif 3565   {csn 3640   <.cotp 3644    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   iota_crio 6297   0gc0g 13400
This theorem is referenced by:  mapdhcl  31917  mapdh6bN  31927  mapdh6cN  31928  mapdh6dN  31929  mapdh8  31979
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-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-ot 3650  df-uni 3828  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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304
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