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Theorem unxpdomlem2 7523
Description: Lemma for unxpdom 7525. (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 >. )
unxpdomlem2.1  |-  ( ph  ->  w  e.  ( a  u.  b ) )
unxpdomlem2.2  |-  ( ph  ->  -.  m  =  n )
unxpdomlem2.3  |-  ( ph  ->  -.  s  =  t )
Assertion
Ref Expression
unxpdomlem2  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  ->  -.  ( F `  z
)  =  ( F `
 w ) )
Distinct variable groups:    w, F, z    a, b, m, n, s, t, w, x, z
Allowed substitution hints:    ph( x, z, w, t, m, n, s, a, b)    F( x, t, m, n, s, a, b)    G( x, z, w, t, m, n, s, a, b)

Proof of Theorem unxpdomlem2
StepHypRef Expression
1 unxpdomlem2.3 . . 3  |-  ( ph  ->  -.  s  =  t )
21adantr 465 . 2  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  ->  -.  s  =  t
)
3 elun1 3528 . . . . . . . . . 10  |-  ( z  e.  a  ->  z  e.  ( a  u.  b
) )
43ad2antrl 727 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  -> 
z  e.  ( a  u.  b ) )
5 unxpdomlem1.1 . . . . . . . . . 10  |-  F  =  ( x  e.  ( a  u.  b ) 
|->  G )
6 unxpdomlem1.2 . . . . . . . . . 10  |-  G  =  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m
) ,  x >. )
75, 6unxpdomlem1 7522 . . . . . . . . 9  |-  ( z  e.  ( a  u.  b )  ->  ( F `  z )  =  if ( z  e.  a ,  <. z ,  if ( z  =  m ,  t ,  s ) >. ,  <. if ( z  =  t ,  n ,  m
) ,  z >.
) )
84, 7syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  -> 
( F `  z
)  =  if ( z  e.  a , 
<. z ,  if ( z  =  m ,  t ,  s )
>. ,  <. if ( z  =  t ,  n ,  m ) ,  z >. )
)
9 iftrue 3802 . . . . . . . . 9  |-  ( z  e.  a  ->  if ( z  e.  a ,  <. z ,  if ( z  =  m ,  t ,  s ) >. ,  <. if ( z  =  t ,  n ,  m ) ,  z >. )  =  <. z ,  if ( z  =  m ,  t ,  s ) >. )
109ad2antrl 727 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  ->  if ( z  e.  a ,  <. z ,  if ( z  =  m ,  t ,  s ) >. ,  <. if ( z  =  t ,  n ,  m ) ,  z >. )  =  <. z ,  if ( z  =  m ,  t ,  s ) >. )
118, 10eqtrd 2475 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  -> 
( F `  z
)  =  <. z ,  if ( z  =  m ,  t ,  s ) >. )
12 unxpdomlem2.1 . . . . . . . . . 10  |-  ( ph  ->  w  e.  ( a  u.  b ) )
1312adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  ->  w  e.  ( a  u.  b ) )
145, 6unxpdomlem1 7522 . . . . . . . . 9  |-  ( w  e.  ( a  u.  b )  ->  ( F `  w )  =  if ( w  e.  a ,  <. w ,  if ( w  =  m ,  t ,  s ) >. ,  <. if ( w  =  t ,  n ,  m
) ,  w >. ) )
1513, 14syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  -> 
( F `  w
)  =  if ( w  e.  a , 
<. w ,  if ( w  =  m ,  t ,  s )
>. ,  <. if ( w  =  t ,  n ,  m ) ,  w >. )
)
16 iffalse 3804 . . . . . . . . 9  |-  ( -.  w  e.  a  ->  if ( w  e.  a ,  <. w ,  if ( w  =  m ,  t ,  s ) >. ,  <. if ( w  =  t ,  n ,  m ) ,  w >. )  =  <. if ( w  =  t ,  n ,  m ) ,  w >. )
1716ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  ->  if ( w  e.  a ,  <. w ,  if ( w  =  m ,  t ,  s ) >. ,  <. if ( w  =  t ,  n ,  m ) ,  w >. )  =  <. if ( w  =  t ,  n ,  m ) ,  w >. )
1815, 17eqtrd 2475 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  -> 
( F `  w
)  =  <. if ( w  =  t ,  n ,  m ) ,  w >. )
1911, 18eqeq12d 2457 . . . . . 6  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  -> 
( ( F `  z )  =  ( F `  w )  <->  <. z ,  if ( z  =  m ,  t ,  s )
>.  =  <. if ( w  =  t ,  n ,  m ) ,  w >. )
)
2019biimpa 484 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  ->  <. z ,  if ( z  =  m ,  t ,  s )
>.  =  <. if ( w  =  t ,  n ,  m ) ,  w >. )
21 vex 2980 . . . . . 6  |-  z  e. 
_V
22 vex 2980 . . . . . . 7  |-  t  e. 
_V
23 vex 2980 . . . . . . 7  |-  s  e. 
_V
2422, 23ifex 3863 . . . . . 6  |-  if ( z  =  m ,  t ,  s )  e.  _V
2521, 24opth 4571 . . . . 5  |-  ( <.
z ,  if ( z  =  m ,  t ,  s )
>.  =  <. if ( w  =  t ,  n ,  m ) ,  w >.  <->  ( z  =  if ( w  =  t ,  n ,  m )  /\  if ( z  =  m ,  t ,  s )  =  w ) )
2620, 25sylib 196 . . . 4  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
( z  =  if ( w  =  t ,  n ,  m
)  /\  if (
z  =  m ,  t ,  s )  =  w ) )
2726simprd 463 . . 3  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  ->  if ( z  =  m ,  t ,  s )  =  w )
28 iftrue 3802 . . . . . . 7  |-  ( z  =  m  ->  if ( z  =  m ,  t ,  s )  =  t )
2927eqeq1d 2451 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
( if ( z  =  m ,  t ,  s )  =  t  <->  w  =  t
) )
3028, 29syl5ib 219 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
( z  =  m  ->  w  =  t ) )
31 iftrue 3802 . . . . . . 7  |-  ( w  =  t  ->  if ( w  =  t ,  n ,  m )  =  n )
3226simpld 459 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
z  =  if ( w  =  t ,  n ,  m ) )
3332eqeq1d 2451 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
( z  =  n  <-> 
if ( w  =  t ,  n ,  m )  =  n ) )
3431, 33syl5ibr 221 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
( w  =  t  ->  z  =  n ) )
3530, 34syld 44 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
( z  =  m  ->  z  =  n ) )
36 unxpdomlem2.2 . . . . . . 7  |-  ( ph  ->  -.  m  =  n )
3736ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  ->  -.  m  =  n
)
38 equequ1 1736 . . . . . . 7  |-  ( z  =  m  ->  (
z  =  n  <->  m  =  n ) )
3938notbid 294 . . . . . 6  |-  ( z  =  m  ->  ( -.  z  =  n  <->  -.  m  =  n ) )
4037, 39syl5ibrcom 222 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
( z  =  m  ->  -.  z  =  n ) )
4135, 40pm2.65d 175 . . . 4  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  ->  -.  z  =  m
)
42 iffalse 3804 . . . 4  |-  ( -.  z  =  m  ->  if ( z  =  m ,  t ,  s )  =  s )
4341, 42syl 16 . . 3  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  ->  if ( z  =  m ,  t ,  s )  =  s )
44 iffalse 3804 . . . . 5  |-  ( -.  w  =  t  ->  if ( w  =  t ,  n ,  m
)  =  m )
4532eqeq1d 2451 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
( z  =  m  <-> 
if ( w  =  t ,  n ,  m )  =  m ) )
4644, 45syl5ibr 221 . . . 4  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
( -.  w  =  t  ->  z  =  m ) )
4741, 46mt3d 125 . . 3  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  ->  w  =  t )
4827, 43, 473eqtr3d 2483 . 2  |-  ( ( ( ph  /\  (
z  e.  a  /\  -.  w  e.  a
) )  /\  ( F `  z )  =  ( F `  w ) )  -> 
s  =  t )
492, 48mtand 659 1  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  ->  -.  ( F `  z
)  =  ( F `
 w ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    u. cun 3331   ifcif 3796   <.cop 3888    e. cmpt 4355   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431
This theorem is referenced by:  unxpdomlem3  7524
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