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Theorem unxpdomlem1 7717
Description: Lemma for unxpdom 7720. (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 1826 . . . 4  |-  ( x  =  z  ->  (
x  e.  a  <->  z  e.  a ) )
3 opeq1 4203 . . . . 5  |-  ( x  =  z  ->  <. x ,  if ( x  =  m ,  t ,  s ) >.  =  <. z ,  if ( x  =  m ,  t ,  s ) >.
)
4 equequ1 1803 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  m  <->  z  =  m ) )
54ifbid 3951 . . . . . 6  |-  ( x  =  z  ->  if ( x  =  m ,  t ,  s )  =  if ( z  =  m ,  t ,  s ) )
65opeq2d 4210 . . . . 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 1803 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  t  <->  z  =  t ) )
98ifbid 3951 . . . . . 6  |-  ( x  =  z  ->  if ( x  =  t ,  n ,  m )  =  if ( z  =  t ,  n ,  m ) )
109opeq1d 4209 . . . . 5  |-  ( x  =  z  ->  <. if ( x  =  t ,  n ,  m ) ,  x >.  =  <. if ( z  =  t ,  n ,  m
) ,  x >. )
11 opeq2 4204 . . . . 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 3956 . . 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 4701 . . 3  |-  <. z ,  if ( z  =  m ,  t ,  s ) >.  e.  _V
17 opex 4701 . . 3  |-  <. if ( z  =  t ,  n ,  m ) ,  z >.  e.  _V
1816, 17ifex 3997 . 2  |-  if ( z  e.  a , 
<. z ,  if ( z  =  m ,  t ,  s )
>. ,  <. if ( z  =  t ,  n ,  m ) ,  z >. )  e.  _V
1914, 15, 18fvmpt 5931 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 1398    e. wcel 1823    u. cun 3459   ifcif 3929   <.cop 4022    |-> cmpt 4497   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578
This theorem is referenced by:  unxpdomlem2  7718  unxpdomlem3  7719
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