MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unxpdomlem1 Structured version   Unicode version

Theorem unxpdomlem1 7629
Description: Lemma for unxpdom 7632. (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 1761 . . . 4  |-  ( x  =  z  ->  (
x  e.  a  <->  z  e.  a ) )
3 opeq1 4168 . . . . 5  |-  ( x  =  z  ->  <. x ,  if ( x  =  m ,  t ,  s ) >.  =  <. z ,  if ( x  =  m ,  t ,  s ) >.
)
4 equequ1 1738 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  m  <->  z  =  m ) )
54ifbid 3920 . . . . . 6  |-  ( x  =  z  ->  if ( x  =  m ,  t ,  s )  =  if ( z  =  m ,  t ,  s ) )
65opeq2d 4175 . . . . 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 1738 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  t  <->  z  =  t ) )
98ifbid 3920 . . . . . 6  |-  ( x  =  z  ->  if ( x  =  t ,  n ,  m )  =  if ( z  =  t ,  n ,  m ) )
109opeq1d 4174 . . . . 5  |-  ( x  =  z  ->  <. if ( x  =  t ,  n ,  m ) ,  x >.  =  <. if ( z  =  t ,  n ,  m
) ,  x >. )
11 opeq2 4169 . . . . 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 3925 . . 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 4665 . . 3  |-  <. z ,  if ( z  =  m ,  t ,  s ) >.  e.  _V
17 opex 4665 . . 3  |-  <. if ( z  =  t ,  n ,  m ) ,  z >.  e.  _V
1816, 17ifex 3967 . 2  |-  if ( z  e.  a , 
<. z ,  if ( z  =  m ,  t ,  s )
>. ,  <. if ( z  =  t ,  n ,  m ) ,  z >. )  e.  _V
1914, 15, 18fvmpt 5884 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 1370    e. wcel 1758    u. cun 3435   ifcif 3900   <.cop 3992    |-> cmpt 4459   ` cfv 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535
This theorem is referenced by:  unxpdomlem2  7630  unxpdomlem3  7631
  Copyright terms: Public domain W3C validator