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Theorem unxpdomlem2 7735
Description: Lemma for unxpdom 7737. (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 3676 . . . . . . . . . 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 7734 . . . . . . . . 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 3950 . . . . . . . . 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 2508 . . . . . . 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 7734 . . . . . . . . 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 3953 . . . . . . . . 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 2508 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  a  /\  -.  w  e.  a ) )  -> 
( F `  w
)  =  <. if ( w  =  t ,  n ,  m ) ,  w >. )
1911, 18eqeq12d 2489 . . . . . 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 3121 . . . . . 6  |-  z  e. 
_V
22 vex 3121 . . . . . . 7  |-  t  e. 
_V
23 vex 3121 . . . . . . 7  |-  s  e. 
_V
2422, 23ifex 4013 . . . . . 6  |-  if ( z  =  m ,  t ,  s )  e.  _V
2521, 24opth 4726 . . . . 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 3950 . . . . . . 7  |-  ( z  =  m  ->  if ( z  =  m ,  t ,  s )  =  t )
2927eqeq1d 2469 . . . . . . 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 3950 . . . . . . 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 2469 . . . . . . 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 1747 . . . . . . 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 3953 . . . 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 3953 . . . . 5  |-  ( -.  w  =  t  ->  if ( w  =  t ,  n ,  m
)  =  m )
4532eqeq1d 2469 . . . . 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 2516 . 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 1379    e. wcel 1767    u. cun 3479   ifcif 3944   <.cop 4038    |-> cmpt 4510   ` cfv 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601
This theorem is referenced by:  unxpdomlem3  7736
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