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Theorem sgplpte22 26241
Description: The predicate "is a degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
sgplpte.1  |-  P  =  (PPoints `  G )
sgplpte.3  |-  S  =  ( seg `  G
)
sgplpte.4  |-  ( ph  ->  G  e. Ibg )
sgplpte.5  |-  ( ph  ->  X  e.  P )
Assertion
Ref Expression
sgplpte22  |-  ( ph  ->  ( X S X )  =  { X } )

Proof of Theorem sgplpte22
Dummy variables  x  f  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sgplpte.3 . . . 4  |-  S  =  ( seg `  G
)
2 df-seg2 26234 . . . . . 6  |-  seg  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f
)  |  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
32a1i 10 . . . . 5  |-  ( ph  ->  seg  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f ) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f )  |  ( z  e.  ( x (btw `  f )
y )  \/  z  =  x  \/  z  =  y ) } ,  { x }
) ) ) )
4 fveq2 5541 . . . . . . 7  |-  ( f  =  G  ->  (PPoints `  f )  =  (PPoints `  G ) )
54adantl 452 . . . . . 6  |-  ( (
ph  /\  f  =  G )  ->  (PPoints `  f )  =  (PPoints `  G ) )
6 fveq2 5541 . . . . . . . . . . . 12  |-  ( f  =  G  ->  (btw `  f )  =  (btw
`  G ) )
76adantl 452 . . . . . . . . . . 11  |-  ( (
ph  /\  f  =  G )  ->  (btw `  f )  =  (btw
`  G ) )
87oveqd 5891 . . . . . . . . . 10  |-  ( (
ph  /\  f  =  G )  ->  (
x (btw `  f
) y )  =  ( x (btw `  G ) y ) )
98eleq2d 2363 . . . . . . . . 9  |-  ( (
ph  /\  f  =  G )  ->  (
z  e.  ( x (btw `  f )
y )  <->  z  e.  ( x (btw `  G ) y ) ) )
10 biidd 228 . . . . . . . . 9  |-  ( (
ph  /\  f  =  G )  ->  (
z  =  x  <->  z  =  x ) )
11 biidd 228 . . . . . . . . 9  |-  ( (
ph  /\  f  =  G )  ->  (
z  =  y  <->  z  =  y ) )
129, 10, 113orbi123d 1251 . . . . . . . 8  |-  ( (
ph  /\  f  =  G )  ->  (
( z  e.  ( x (btw `  f
) y )  \/  z  =  x  \/  z  =  y )  <-> 
( z  e.  ( x (btw `  G
) y )  \/  z  =  x  \/  z  =  y ) ) )
135, 12rabeqbidv 2796 . . . . . . 7  |-  ( (
ph  /\  f  =  G )  ->  { z  e.  (PPoints `  f
)  |  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y ) }  =  { z  e.  (PPoints `  G )  |  ( z  e.  ( x (btw `  G )
y )  \/  z  =  x  \/  z  =  y ) } )
1413ifeq1d 3592 . . . . . 6  |-  ( (
ph  /\  f  =  G )  ->  if ( x  =/=  y ,  { z  e.  (PPoints `  f )  |  ( z  e.  ( x (btw `  f )
y )  \/  z  =  x  \/  z  =  y ) } ,  { x }
)  =  if ( x  =/=  y ,  { z  e.  (PPoints `  G )  |  ( z  e.  ( x (btw `  G )
y )  \/  z  =  x  \/  z  =  y ) } ,  { x }
) )
155, 5, 14mpt2eq123dv 5926 . . . . 5  |-  ( (
ph  /\  f  =  G )  ->  (
x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f
)  |  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) )  =  ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
16 sgplpte.4 . . . . 5  |-  ( ph  ->  G  e. Ibg )
17 fvex 5555 . . . . . . 7  |-  (PPoints `  G
)  e.  _V
1817, 17mpt2ex 6214 . . . . . 6  |-  ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) )  e.  _V
1918a1i 10 . . . . 5  |-  ( ph  ->  ( x  e.  (PPoints `  G ) ,  y  e.  (PPoints `  G
)  |->  if ( x  =/=  y ,  {
z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) )  e.  _V )
203, 15, 16, 19fvmptd 5622 . . . 4  |-  ( ph  ->  ( seg `  G
)  =  ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
211, 20syl5eq 2340 . . 3  |-  ( ph  ->  S  =  ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
2221oveqd 5891 . 2  |-  ( ph  ->  ( X S X )  =  ( X ( x  e.  (PPoints `  G ) ,  y  e.  (PPoints `  G
)  |->  if ( x  =/=  y ,  {
z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) X ) )
23 sgplpte.1 . . . . . 6  |-  P  =  (PPoints `  G )
2423eqcomi 2300 . . . . 5  |-  (PPoints `  G
)  =  P
2524a1i 10 . . . 4  |-  ( ph  ->  (PPoints `  G )  =  P )
26 biidd 228 . . . . . 6  |-  ( ph  ->  ( ( z  e.  ( x (btw `  G ) y )  \/  z  =  x  \/  z  =  y )  <->  ( z  e.  ( x (btw `  G ) y )  \/  z  =  x  \/  z  =  y ) ) )
2725, 26rabeqbidv 2796 . . . . 5  |-  ( ph  ->  { z  e.  (PPoints `  G )  |  ( z  e.  ( x (btw `  G )
y )  \/  z  =  x  \/  z  =  y ) }  =  { z  e.  P  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } )
2827ifeq1d 3592 . . . 4  |-  ( ph  ->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } )  =  if ( x  =/=  y ,  {
z  e.  P  | 
( z  e.  ( x (btw `  G
) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) )
2925, 25, 28mpt2eq123dv 5926 . . 3  |-  ( ph  ->  ( x  e.  (PPoints `  G ) ,  y  e.  (PPoints `  G
)  |->  if ( x  =/=  y ,  {
z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) )  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  { z  e.  P  |  ( z  e.  ( x (btw `  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  {
x } ) ) )
30 simpl 443 . . . . . 6  |-  ( ( x  =  X  /\  y  =  X )  ->  x  =  X )
31 simpr 447 . . . . . 6  |-  ( ( x  =  X  /\  y  =  X )  ->  y  =  X )
3230, 31neeq12d 2474 . . . . 5  |-  ( ( x  =  X  /\  y  =  X )  ->  ( x  =/=  y  <->  X  =/=  X ) )
3332adantl 452 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  X ) )  -> 
( x  =/=  y  <->  X  =/=  X ) )
34 oveq12 5883 . . . . . . . 8  |-  ( ( x  =  X  /\  y  =  X )  ->  ( x (btw `  G ) y )  =  ( X (btw
`  G ) X ) )
3534eleq2d 2363 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  X )  ->  ( z  e.  ( x (btw `  G
) y )  <->  z  e.  ( X (btw `  G
) X ) ) )
3635adantl 452 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  X ) )  -> 
( z  e.  ( x (btw `  G
) y )  <->  z  e.  ( X (btw `  G
) X ) ) )
37 eqeq2 2305 . . . . . . 7  |-  ( x  =  X  ->  (
z  =  x  <->  z  =  X ) )
3837ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  X ) )  -> 
( z  =  x  <-> 
z  =  X ) )
39 eqeq2 2305 . . . . . . . 8  |-  ( y  =  X  ->  (
z  =  y  <->  z  =  X ) )
4039adantl 452 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  X )  ->  ( z  =  y  <-> 
z  =  X ) )
4140adantl 452 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  X ) )  -> 
( z  =  y  <-> 
z  =  X ) )
4236, 38, 413orbi123d 1251 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  X ) )  -> 
( ( z  e.  ( x (btw `  G ) y )  \/  z  =  x  \/  z  =  y )  <->  ( z  e.  ( X (btw `  G ) X )  \/  z  =  X  \/  z  =  X ) ) )
4342rabbidv 2793 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  X ) )  ->  { z  e.  P  |  ( z  e.  ( x (btw `  G ) y )  \/  z  =  x  \/  z  =  y ) }  =  {
z  e.  P  | 
( z  e.  ( X (btw `  G
) X )  \/  z  =  X  \/  z  =  X ) } )
44 sneq 3664 . . . . 5  |-  ( x  =  X  ->  { x }  =  { X } )
4544ad2antrl 708 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  X ) )  ->  { x }  =  { X } )
4633, 43, 45ifbieq12d 3600 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  X ) )  ->  if ( x  =/=  y ,  { z  e.  P  |  ( z  e.  ( x (btw `  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  {
x } )  =  if ( X  =/= 
X ,  { z  e.  P  |  ( z  e.  ( X (btw `  G ) X )  \/  z  =  X  \/  z  =  X ) } ,  { X } ) )
47 sgplpte.5 . . 3  |-  ( ph  ->  X  e.  P )
4823, 17eqeltri 2366 . . . . . 6  |-  P  e. 
_V
4948rabex 4181 . . . . 5  |-  { z  e.  P  |  ( z  e.  ( X (btw `  G ) X )  \/  z  =  X  \/  z  =  X ) }  e.  _V
50 snex 4232 . . . . 5  |-  { X }  e.  _V
5149, 50ifex 3636 . . . 4  |-  if ( X  =/=  X ,  { z  e.  P  |  ( z  e.  ( X (btw `  G ) X )  \/  z  =  X  \/  z  =  X ) } ,  { X } )  e.  _V
5251a1i 10 . . 3  |-  ( ph  ->  if ( X  =/= 
X ,  { z  e.  P  |  ( z  e.  ( X (btw `  G ) X )  \/  z  =  X  \/  z  =  X ) } ,  { X } )  e. 
_V )
5329, 46, 47, 47, 52ovmpt2d 5991 . 2  |-  ( ph  ->  ( X ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) X )  =  if ( X  =/= 
X ,  { z  e.  P  |  ( z  e.  ( X (btw `  G ) X )  \/  z  =  X  \/  z  =  X ) } ,  { X } ) )
54 neirr 2464 . . 3  |-  -.  X  =/=  X
55 iffalse 3585 . . 3  |-  ( -.  X  =/=  X  ->  if ( X  =/=  X ,  { z  e.  P  |  ( z  e.  ( X (btw `  G ) X )  \/  z  =  X  \/  z  =  X ) } ,  { X } )  =  { X } )
5654, 55mp1i 11 . 2  |-  ( ph  ->  if ( X  =/= 
X ,  { z  e.  P  |  ( z  e.  ( X (btw `  G ) X )  \/  z  =  X  \/  z  =  X ) } ,  { X } )  =  { X } )
5722, 53, 563eqtrd 2332 1  |-  ( ph  ->  ( X S X )  =  { X } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801   ifcif 3578   {csn 3653    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876  PPointscpoints 26159  btwcbtw 26209  Ibgcibg 26210   segcseg 26233
This theorem is referenced by:  sgplpte21d1  26242  segline  26244  bsstrs  26249  nbssntrs  26250  segray  26258  bosser  26270  pdiveql  26271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-seg2 26234
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