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Theorem hdmap1cbv 31993
Description: Frequently used lemma to change bound variables in  L hypothesis. (Contributed by NM, 15-May-2015.)
Hypothesis
Ref Expression
hdmap1cbv.l  |-  L  =  ( 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 ) } ) ) ) ) )
Assertion
Ref Expression
hdmap1cbv  |-  L  =  ( y  e.  _V  |->  if ( ( 2nd `  y
)  =  .0.  ,  Q ,  ( iota_ i  e.  D ( ( M `  ( N `
 { ( 2nd `  y ) } ) )  =  ( J `
 { i } )  /\  ( M `
 ( N `  { ( ( 1st `  ( 1st `  y
) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  {
( ( 2nd `  ( 1st `  y ) ) R i ) } ) ) ) ) )
Distinct variable groups:    h, i, x, y, D    h, J, i, x, y    h, M, i, x, y    h, N, i, x, y    x,  .0. , y    x, Q, y    R, h, i, x, y    .- , h, i, x, y
Allowed substitution hints:    Q( h, i)    L( x, y, h, i)    .0. ( h, i)

Proof of Theorem hdmap1cbv
StepHypRef Expression
1 hdmap1cbv.l . 2  |-  L  =  ( 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 ) } ) ) ) ) )
2 fveq2 5525 . . . . 5  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
32eqeq1d 2291 . . . 4  |-  ( x  =  y  ->  (
( 2nd `  x
)  =  .0.  <->  ( 2nd `  y )  =  .0.  ) )
42sneqd 3653 . . . . . . . . 9  |-  ( x  =  y  ->  { ( 2nd `  x ) }  =  { ( 2nd `  y ) } )
54fveq2d 5529 . . . . . . . 8  |-  ( x  =  y  ->  ( N `  { ( 2nd `  x ) } )  =  ( N `
 { ( 2nd `  y ) } ) )
65fveq2d 5529 . . . . . . 7  |-  ( x  =  y  ->  ( M `  ( N `  { ( 2nd `  x
) } ) )  =  ( M `  ( N `  { ( 2nd `  y ) } ) ) )
76eqeq1d 2291 . . . . . 6  |-  ( x  =  y  ->  (
( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  { h } )  <->  ( M `  ( N `  {
( 2nd `  y
) } ) )  =  ( J `  { h } ) ) )
8 fveq2 5525 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
98fveq2d 5529 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( 1st `  ( 1st `  x
) )  =  ( 1st `  ( 1st `  y ) ) )
109, 2oveq12d 5876 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) )  =  ( ( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) )
1110sneqd 3653 . . . . . . . . 9  |-  ( x  =  y  ->  { ( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) }  =  { ( ( 1st `  ( 1st `  y
) )  .-  ( 2nd `  y ) ) } )
1211fveq2d 5529 . . . . . . . 8  |-  ( x  =  y  ->  ( N `  { (
( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } )  =  ( N `  { ( ( 1st `  ( 1st `  y
) )  .-  ( 2nd `  y ) ) } ) )
1312fveq2d 5529 . . . . . . 7  |-  ( x  =  y  ->  ( M `  ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) )  =  ( M `  ( N `  { (
( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) } ) ) )
148fveq2d 5529 . . . . . . . . . 10  |-  ( x  =  y  ->  ( 2nd `  ( 1st `  x
) )  =  ( 2nd `  ( 1st `  y ) ) )
1514oveq1d 5873 . . . . . . . . 9  |-  ( x  =  y  ->  (
( 2nd `  ( 1st `  x ) ) R h )  =  ( ( 2nd `  ( 1st `  y ) ) R h ) )
1615sneqd 3653 . . . . . . . 8  |-  ( x  =  y  ->  { ( ( 2nd `  ( 1st `  x ) ) R h ) }  =  { ( ( 2nd `  ( 1st `  y ) ) R h ) } )
1716fveq2d 5529 . . . . . . 7  |-  ( x  =  y  ->  ( J `  { (
( 2nd `  ( 1st `  x ) ) R h ) } )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R h ) } ) )
1813, 17eqeq12d 2297 . . . . . 6  |-  ( x  =  y  ->  (
( M `  ( N `  { (
( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } )  <-> 
( M `  ( N `  { (
( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R h ) } ) ) )
197, 18anbi12d 691 . . . . 5  |-  ( x  =  y  ->  (
( ( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  x
) ) R h ) } ) )  <-> 
( ( M `  ( N `  { ( 2nd `  y ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  y ) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  y
) ) R h ) } ) ) ) )
2019riotabidv 6306 . . . 4  |-  ( x  =  y  ->  ( iota_ h  e.  D ( ( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) )  =  ( J `  {
( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) )  =  ( iota_ h  e.  D
( ( M `  ( N `  { ( 2nd `  y ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  y ) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  y
) ) R h ) } ) ) ) )
213, 20ifbieq2d 3585 . . 3  |-  ( x  =  y  ->  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 ) } ) ) ) )  =  if ( ( 2nd `  y )  =  .0. 
,  Q ,  (
iota_ h  e.  D
( ( M `  ( N `  { ( 2nd `  y ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  y ) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  y
) ) R h ) } ) ) ) ) )
2221cbvmptv 4111 . 2  |-  ( 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 ) } ) ) ) ) )  =  ( y  e. 
_V  |->  if ( ( 2nd `  y )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `
 ( N `  { ( 2nd `  y
) } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( ( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R h ) } ) ) ) ) )
23 sneq 3651 . . . . . . . 8  |-  ( h  =  i  ->  { h }  =  { i } )
2423fveq2d 5529 . . . . . . 7  |-  ( h  =  i  ->  ( J `  { h } )  =  ( J `  { i } ) )
2524eqeq2d 2294 . . . . . 6  |-  ( h  =  i  ->  (
( M `  ( N `  { ( 2nd `  y ) } ) )  =  ( J `  { h } )  <->  ( M `  ( N `  {
( 2nd `  y
) } ) )  =  ( J `  { i } ) ) )
26 oveq2 5866 . . . . . . . . 9  |-  ( h  =  i  ->  (
( 2nd `  ( 1st `  y ) ) R h )  =  ( ( 2nd `  ( 1st `  y ) ) R i ) )
2726sneqd 3653 . . . . . . . 8  |-  ( h  =  i  ->  { ( ( 2nd `  ( 1st `  y ) ) R h ) }  =  { ( ( 2nd `  ( 1st `  y ) ) R i ) } )
2827fveq2d 5529 . . . . . . 7  |-  ( h  =  i  ->  ( J `  { (
( 2nd `  ( 1st `  y ) ) R h ) } )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R i ) } ) )
2928eqeq2d 2294 . . . . . 6  |-  ( h  =  i  ->  (
( M `  ( N `  { (
( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R h ) } )  <-> 
( M `  ( N `  { (
( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R i ) } ) ) )
3025, 29anbi12d 691 . . . . 5  |-  ( h  =  i  ->  (
( ( M `  ( N `  { ( 2nd `  y ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  y ) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  y
) ) R h ) } ) )  <-> 
( ( M `  ( N `  { ( 2nd `  y ) } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  y ) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  y
) ) R i ) } ) ) ) )
3130cbvriotav 6316 . . . 4  |-  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  y ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R h ) } ) ) )  =  (
iota_ i  e.  D
( ( M `  ( N `  { ( 2nd `  y ) } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  y ) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  y
) ) R i ) } ) ) )
32 ifeq2 3570 . . . 4  |-  ( (
iota_ h  e.  D
( ( M `  ( N `  { ( 2nd `  y ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  y ) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  y
) ) R h ) } ) ) )  =  ( iota_ i  e.  D ( ( M `  ( N `
 { ( 2nd `  y ) } ) )  =  ( J `
 { i } )  /\  ( M `
 ( N `  { ( ( 1st `  ( 1st `  y
) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  {
( ( 2nd `  ( 1st `  y ) ) R i ) } ) ) )  ->  if ( ( 2nd `  y
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  y ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R h ) } ) ) ) )  =  if ( ( 2nd `  y )  =  .0. 
,  Q ,  (
iota_ i  e.  D
( ( M `  ( N `  { ( 2nd `  y ) } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  y ) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  y
) ) R i ) } ) ) ) ) )
3331, 32ax-mp 8 . . 3  |-  if ( ( 2nd `  y
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  y ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R h ) } ) ) ) )  =  if ( ( 2nd `  y )  =  .0. 
,  Q ,  (
iota_ i  e.  D
( ( M `  ( N `  { ( 2nd `  y ) } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  y ) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  y
) ) R i ) } ) ) ) )
3433mpteq2i 4103 . 2  |-  ( y  e.  _V  |->  if ( ( 2nd `  y
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  y ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R h ) } ) ) ) ) )  =  ( y  e. 
_V  |->  if ( ( 2nd `  y )  =  .0.  ,  Q ,  ( iota_ i  e.  D ( ( M `
 ( N `  { ( 2nd `  y
) } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( ( 1st `  ( 1st `  y ) ) 
.-  ( 2nd `  y
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  y ) ) R i ) } ) ) ) ) )
351, 22, 343eqtri 2307 1  |-  L  =  ( y  e.  _V  |->  if ( ( 2nd `  y
)  =  .0.  ,  Q ,  ( iota_ i  e.  D ( ( M `  ( N `
 { ( 2nd `  y ) } ) )  =  ( J `
 { i } )  /\  ( M `
 ( N `  { ( ( 1st `  ( 1st `  y
) )  .-  ( 2nd `  y ) ) } ) )  =  ( J `  {
( ( 2nd `  ( 1st `  y ) ) R i ) } ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623   _Vcvv 2788   ifcif 3565   {csn 3640    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   iota_crio 6297
This theorem is referenced by:  hdmap1valc  31994  hdmap1eu  32016  hdmap1euOLDN  32017
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  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-br 4024  df-opab 4078  df-mpt 4079  df-iota 5219  df-fv 5263  df-ov 5861  df-riota 6304
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