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Theorem isray2 26256
Description: A degenerated ray. (For my private use only. Don't use.) (Contributed by FL, 13-Apr-2016.)
Hypotheses
Ref Expression
isray2.1  |-  P  =  (PPoints `  G )
isray2.3  |-  R  =  (ray `  G )
isray2.5  |-  ( ph  ->  G  e. Ibg )
isray2.6  |-  ( ph  ->  X  e.  P )
Assertion
Ref Expression
isray2  |-  ( ph  ->  ( X R X )  =  { X } )

Proof of Theorem isray2
Dummy variables  x  f  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isray2.3 . . . 4  |-  R  =  (ray `  G )
2 df-ray2 26255 . . . . . 6  |- ray  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  ( ( x ( seg `  f
) y )  u. 
{ z  e.  (PPoints `  f )  |  y  e.  ( x (btw
`  f ) z ) } ) ,  { x } ) ) )
32a1i 10 . . . . 5  |-  ( ph  -> ray  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f ) ,  y  e.  (PPoints `  f
)  |->  if ( x  =/=  y ,  ( ( x ( seg `  f ) y )  u.  { z  e.  (PPoints `  f )  |  y  e.  (
x (btw `  f
) z ) } ) ,  { x } ) ) ) )
4 fveq2 5541 . . . . . . . 8  |-  ( f  =  G  ->  (PPoints `  f )  =  (PPoints `  G ) )
5 isray2.1 . . . . . . . 8  |-  P  =  (PPoints `  G )
64, 5syl6eqr 2346 . . . . . . 7  |-  ( f  =  G  ->  (PPoints `  f )  =  P )
76adantl 452 . . . . . 6  |-  ( (
ph  /\  f  =  G )  ->  (PPoints `  f )  =  P )
8 fveq2 5541 . . . . . . . . . 10  |-  ( f  =  G  ->  ( seg `  f )  =  ( seg `  G
) )
98oveqd 5891 . . . . . . . . 9  |-  ( f  =  G  ->  (
x ( seg `  f
) y )  =  ( x ( seg `  G ) y ) )
10 fveq2 5541 . . . . . . . . . . . 12  |-  ( f  =  G  ->  (btw `  f )  =  (btw
`  G ) )
1110oveqd 5891 . . . . . . . . . . 11  |-  ( f  =  G  ->  (
x (btw `  f
) z )  =  ( x (btw `  G ) z ) )
1211eleq2d 2363 . . . . . . . . . 10  |-  ( f  =  G  ->  (
y  e.  ( x (btw `  f )
z )  <->  y  e.  ( x (btw `  G ) z ) ) )
134, 12rabeqbidv 2796 . . . . . . . . 9  |-  ( f  =  G  ->  { z  e.  (PPoints `  f
)  |  y  e.  ( x (btw `  f ) z ) }  =  { z  e.  (PPoints `  G
)  |  y  e.  ( x (btw `  G ) z ) } )
149, 13uneq12d 3343 . . . . . . . 8  |-  ( f  =  G  ->  (
( x ( seg `  f ) y )  u.  { z  e.  (PPoints `  f )  |  y  e.  (
x (btw `  f
) z ) } )  =  ( ( x ( seg `  G
) y )  u. 
{ z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) } ) )
1514ifeq1d 3592 . . . . . . 7  |-  ( f  =  G  ->  if ( x  =/=  y ,  ( ( x ( seg `  f
) y )  u. 
{ z  e.  (PPoints `  f )  |  y  e.  ( x (btw
`  f ) z ) } ) ,  { x } )  =  if ( x  =/=  y ,  ( ( x ( seg `  G ) y )  u.  { z  e.  (PPoints `  G )  |  y  e.  (
x (btw `  G
) z ) } ) ,  { x } ) )
1615adantl 452 . . . . . 6  |-  ( (
ph  /\  f  =  G )  ->  if ( x  =/=  y ,  ( ( x ( seg `  f
) y )  u. 
{ z  e.  (PPoints `  f )  |  y  e.  ( x (btw
`  f ) z ) } ) ,  { x } )  =  if ( x  =/=  y ,  ( ( x ( seg `  G ) y )  u.  { z  e.  (PPoints `  G )  |  y  e.  (
x (btw `  G
) z ) } ) ,  { x } ) )
177, 7, 16mpt2eq123dv 5926 . . . . 5  |-  ( (
ph  /\  f  =  G )  ->  (
x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  ( ( x ( seg `  f
) y )  u. 
{ z  e.  (PPoints `  f )  |  y  e.  ( x (btw
`  f ) z ) } ) ,  { x } ) )  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( ( x ( seg `  G ) y )  u.  {
z  e.  (PPoints `  G
)  |  y  e.  ( x (btw `  G ) z ) } ) ,  {
x } ) ) )
18 isray2.5 . . . . 5  |-  ( ph  ->  G  e. Ibg )
19 fvex 5555 . . . . . . . 8  |-  (PPoints `  G
)  e.  _V
205, 19eqeltri 2366 . . . . . . 7  |-  P  e. 
_V
2120, 20mpt2ex 6214 . . . . . 6  |-  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( ( x ( seg `  G ) y )  u.  {
z  e.  (PPoints `  G
)  |  y  e.  ( x (btw `  G ) z ) } ) ,  {
x } ) )  e.  _V
2221a1i 10 . . . . 5  |-  ( ph  ->  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( ( x ( seg `  G
) y )  u. 
{ z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) } ) ,  { x } ) )  e.  _V )
233, 17, 18, 22fvmptd 5622 . . . 4  |-  ( ph  ->  (ray `  G )  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( ( x ( seg `  G ) y )  u.  { z  e.  (PPoints `  G )  |  y  e.  (
x (btw `  G
) z ) } ) ,  { x } ) ) )
241, 23syl5eq 2340 . . 3  |-  ( ph  ->  R  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( ( x ( seg `  G ) y )  u.  {
z  e.  (PPoints `  G
)  |  y  e.  ( x (btw `  G ) z ) } ) ,  {
x } ) ) )
25 simpl 443 . . . . . 6  |-  ( ( x  =  X  /\  y  =  X )  ->  x  =  X )
26 simpr 447 . . . . . 6  |-  ( ( x  =  X  /\  y  =  X )  ->  y  =  X )
2725, 26neeq12d 2474 . . . . 5  |-  ( ( x  =  X  /\  y  =  X )  ->  ( x  =/=  y  <->  X  =/=  X ) )
28 oveq12 5883 . . . . . 6  |-  ( ( x  =  X  /\  y  =  X )  ->  ( x ( seg `  G ) y )  =  ( X ( seg `  G ) X ) )
29 oveq1 5881 . . . . . . . . 9  |-  ( x  =  X  ->  (
x (btw `  G
) z )  =  ( X (btw `  G ) z ) )
3029adantr 451 . . . . . . . 8  |-  ( ( x  =  X  /\  y  =  X )  ->  ( x (btw `  G ) z )  =  ( X (btw
`  G ) z ) )
3126, 30eleq12d 2364 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  X )  ->  ( y  e.  ( x (btw `  G
) z )  <->  X  e.  ( X (btw `  G
) z ) ) )
3231rabbidv 2793 . . . . . 6  |-  ( ( x  =  X  /\  y  =  X )  ->  { z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) }  =  {
z  e.  (PPoints `  G
)  |  X  e.  ( X (btw `  G ) z ) } )
3328, 32uneq12d 3343 . . . . 5  |-  ( ( x  =  X  /\  y  =  X )  ->  ( ( x ( seg `  G ) y )  u.  {
z  e.  (PPoints `  G
)  |  y  e.  ( x (btw `  G ) z ) } )  =  ( ( X ( seg `  G ) X )  u.  { z  e.  (PPoints `  G )  |  X  e.  ( X (btw `  G )
z ) } ) )
34 sneq 3664 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
3534adantr 451 . . . . 5  |-  ( ( x  =  X  /\  y  =  X )  ->  { x }  =  { X } )
3627, 33, 35ifbieq12d 3600 . . . 4  |-  ( ( x  =  X  /\  y  =  X )  ->  if ( x  =/=  y ,  ( ( x ( seg `  G
) y )  u. 
{ z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) } ) ,  { x } )  =  if ( X  =/=  X ,  ( ( X ( seg `  G ) X )  u.  { z  e.  (PPoints `  G )  |  X  e.  ( X (btw `  G )
z ) } ) ,  { X }
) )
3736adantl 452 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  X ) )  ->  if ( x  =/=  y ,  ( ( x ( seg `  G
) y )  u. 
{ z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) } ) ,  { x } )  =  if ( X  =/=  X ,  ( ( X ( seg `  G ) X )  u.  { z  e.  (PPoints `  G )  |  X  e.  ( X (btw `  G )
z ) } ) ,  { X }
) )
38 isray2.6 . . 3  |-  ( ph  ->  X  e.  P )
39 ovex 5899 . . . . . 6  |-  ( X ( seg `  G
) X )  e. 
_V
4019rabex 4181 . . . . . 6  |-  { z  e.  (PPoints `  G
)  |  X  e.  ( X (btw `  G ) z ) }  e.  _V
4139, 40unex 4534 . . . . 5  |-  ( ( X ( seg `  G
) X )  u. 
{ z  e.  (PPoints `  G )  |  X  e.  ( X (btw `  G ) z ) } )  e.  _V
42 snex 4232 . . . . 5  |-  { X }  e.  _V
4341, 42ifex 3636 . . . 4  |-  if ( X  =/=  X , 
( ( X ( seg `  G ) X )  u.  {
z  e.  (PPoints `  G
)  |  X  e.  ( X (btw `  G ) z ) } ) ,  { X } )  e.  _V
4443a1i 10 . . 3  |-  ( ph  ->  if ( X  =/= 
X ,  ( ( X ( seg `  G
) X )  u. 
{ z  e.  (PPoints `  G )  |  X  e.  ( X (btw `  G ) z ) } ) ,  { X } )  e.  _V )
4524, 37, 38, 38, 44ovmpt2d 5991 . 2  |-  ( ph  ->  ( X R X )  =  if ( X  =/=  X , 
( ( X ( seg `  G ) X )  u.  {
z  e.  (PPoints `  G
)  |  X  e.  ( X (btw `  G ) z ) } ) ,  { X } ) )
46 neirr 2464 . . 3  |-  -.  X  =/=  X
47 iffalse 3585 . . 3  |-  ( -.  X  =/=  X  ->  if ( X  =/=  X ,  ( ( X ( seg `  G
) X )  u. 
{ z  e.  (PPoints `  G )  |  X  e.  ( X (btw `  G ) z ) } ) ,  { X } )  =  { X } )
4846, 47mp1i 11 . 2  |-  ( ph  ->  if ( X  =/= 
X ,  ( ( X ( seg `  G
) X )  u. 
{ z  e.  (PPoints `  G )  |  X  e.  ( X (btw `  G ) z ) } ) ,  { X } )  =  { X } )
4945, 48eqtrd 2328 1  |-  ( ph  ->  ( X R X )  =  { X } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    u. cun 3163   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  raycray2 26254
This theorem is referenced by:  segray  26258  rayline  26259
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-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-ray2 26255
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