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Theorem unxpdomlem1 7801
Description: Lemma for unxpdom 7804. (Trivial substitution proof.) (Contributed by Mario Carneiro, 13-Jan-2013.)
Hypotheses
Ref Expression
unxpdomlem1.1  |-  F  =  ( x  e.  ( a  u.  b ) 
|->  G )
unxpdomlem1.2  |-  G  =  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m
) ,  x >. )
Assertion
Ref Expression
unxpdomlem1  |-  ( z  e.  ( a  u.  b )  ->  ( F `  z )  =  if ( z  e.  a ,  <. z ,  if ( z  =  m ,  t ,  s ) >. ,  <. if ( z  =  t ,  n ,  m
) ,  z >.
) )
Distinct variable groups:    z, F    a, b, m, n, s, t, x, z
Allowed substitution hints:    F( x, t, m, n, s, a, b)    G( x, z, t, m, n, s, a, b)

Proof of Theorem unxpdomlem1
StepHypRef Expression
1 unxpdomlem1.2 . . 3  |-  G  =  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m
) ,  x >. )
2 elequ1 1904 . . . 4  |-  ( x  =  z  ->  (
x  e.  a  <->  z  e.  a ) )
3 opeq1 4179 . . . . 5  |-  ( x  =  z  ->  <. x ,  if ( x  =  m ,  t ,  s ) >.  =  <. z ,  if ( x  =  m ,  t ,  s ) >.
)
4 equequ1 1877 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  m  <->  z  =  m ) )
54ifbid 3914 . . . . . 6  |-  ( x  =  z  ->  if ( x  =  m ,  t ,  s )  =  if ( z  =  m ,  t ,  s ) )
65opeq2d 4186 . . . . 5  |-  ( x  =  z  ->  <. z ,  if ( x  =  m ,  t ,  s ) >.  =  <. z ,  if ( z  =  m ,  t ,  s ) >.
)
73, 6eqtrd 2495 . . . 4  |-  ( x  =  z  ->  <. x ,  if ( x  =  m ,  t ,  s ) >.  =  <. z ,  if ( z  =  m ,  t ,  s ) >.
)
8 equequ1 1877 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  t  <->  z  =  t ) )
98ifbid 3914 . . . . . 6  |-  ( x  =  z  ->  if ( x  =  t ,  n ,  m )  =  if ( z  =  t ,  n ,  m ) )
109opeq1d 4185 . . . . 5  |-  ( x  =  z  ->  <. if ( x  =  t ,  n ,  m ) ,  x >.  =  <. if ( z  =  t ,  n ,  m
) ,  x >. )
11 opeq2 4180 . . . . 5  |-  ( x  =  z  ->  <. if ( z  =  t ,  n ,  m ) ,  x >.  =  <. if ( z  =  t ,  n ,  m
) ,  z >.
)
1210, 11eqtrd 2495 . . . 4  |-  ( x  =  z  ->  <. if ( x  =  t ,  n ,  m ) ,  x >.  =  <. if ( z  =  t ,  n ,  m
) ,  z >.
)
132, 7, 12ifbieq12d 3919 . . 3  |-  ( x  =  z  ->  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m ) ,  x >. )  =  if ( z  e.  a ,  <. z ,  if ( z  =  m ,  t ,  s ) >. ,  <. if ( z  =  t ,  n ,  m
) ,  z >.
) )
141, 13syl5eq 2507 . 2  |-  ( x  =  z  ->  G  =  if ( z  e.  a ,  <. z ,  if ( z  =  m ,  t ,  s ) >. ,  <. if ( z  =  t ,  n ,  m
) ,  z >.
) )
15 unxpdomlem1.1 . 2  |-  F  =  ( x  e.  ( a  u.  b ) 
|->  G )
16 opex 4677 . . 3  |-  <. z ,  if ( z  =  m ,  t ,  s ) >.  e.  _V
17 opex 4677 . . 3  |-  <. if ( z  =  t ,  n ,  m ) ,  z >.  e.  _V
1816, 17ifex 3960 . 2  |-  if ( z  e.  a , 
<. z ,  if ( z  =  m ,  t ,  s )
>. ,  <. if ( z  =  t ,  n ,  m ) ,  z >. )  e.  _V
1914, 15, 18fvmpt 5970 1  |-  ( z  e.  ( a  u.  b )  ->  ( F `  z )  =  if ( z  e.  a ,  <. z ,  if ( z  =  m ,  t ,  s ) >. ,  <. if ( z  =  t ,  n ,  m
) ,  z >.
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454    e. wcel 1897    u. cun 3413   ifcif 3892   <.cop 3985    |-> cmpt 4474   ` cfv 5600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5564  df-fun 5602  df-fv 5608
This theorem is referenced by:  unxpdomlem2  7802  unxpdomlem3  7803
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