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Theorem fvray 24764
Description: Calculate the value of the Ray function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fvray  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  -> 
( PRay A )  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } )
Distinct variable groups:    x, A    x, N    x, P

Proof of Theorem fvray
Dummy variables  a  n  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 5861 . 2  |-  ( PRay A )  =  (Ray
`  <. P ,  A >. )
2 eqid 2283 . . . . 5  |-  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }
3 fveq2 5525 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
43eleq2d 2350 . . . . . . . 8  |-  ( n  =  N  ->  ( P  e.  ( EE `  n )  <->  P  e.  ( EE `  N ) ) )
53eleq2d 2350 . . . . . . . 8  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
64, 53anbi12d 1253 . . . . . . 7  |-  ( n  =  N  ->  (
( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  <->  ( P  e.  ( EE `  N
)  /\  A  e.  ( EE `  N )  /\  P  =/=  A
) ) )
7 rabeq 2782 . . . . . . . . 9  |-  ( ( EE `  n )  =  ( EE `  N )  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } )
83, 7syl 15 . . . . . . . 8  |-  ( n  =  N  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } )
98eqeq2d 2294 . . . . . . 7  |-  ( n  =  N  ->  ( { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. }  =  { x  e.  ( EE `  n
)  |  POutsideOf <. A ,  x >. }  <->  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } ) )
106, 9anbi12d 691 . . . . . 6  |-  ( n  =  N  ->  (
( ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } )  <->  ( ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } ) ) )
1110rspcev 2884 . . . . 5  |-  ( ( N  e.  NN  /\  ( ( P  e.  ( EE `  N
)  /\  A  e.  ( EE `  N )  /\  P  =/=  A
)  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } ) )  ->  E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) )
122, 11mpanr2 665 . . . 4  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) )
13 simpr1 961 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  P  e.  ( EE `  N ) )
14 simpr2 962 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  A  e.  ( EE `  N ) )
15 fvex 5539 . . . . . . 7  |-  ( EE
`  N )  e. 
_V
1615rabex 4165 . . . . . 6  |-  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  e.  _V
17 eleq1 2343 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p  e.  ( EE
`  n )  <->  P  e.  ( EE `  n ) ) )
18 neeq1 2454 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p  =/=  a  <->  P  =/=  a ) )
1917, 183anbi13d 1254 . . . . . . . . 9  |-  ( p  =  P  ->  (
( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  <->  ( P  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  P  =/=  a
) ) )
20 breq1 4026 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
pOutsideOf <. a ,  x >.  <-> 
POutsideOf <. a ,  x >. ) )
2120rabbidv 2780 . . . . . . . . . 10  |-  ( p  =  P  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. a ,  x >. } )
2221eqeq2d 2294 . . . . . . . . 9  |-  ( p  =  P  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } ) )
2319, 22anbi12d 691 . . . . . . . 8  |-  ( p  =  P  ->  (
( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  <->  ( ( P  e.  ( EE `  n )  /\  a  e.  ( EE `  n
)  /\  P  =/=  a )  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } ) ) )
2423rexbidv 2564 . . . . . . 7  |-  ( p  =  P  ->  ( E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  a  e.  ( EE `  n
)  /\  P  =/=  a )  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } ) ) )
25 eleq1 2343 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
26 neeq2 2455 . . . . . . . . . 10  |-  ( a  =  A  ->  ( P  =/=  a  <->  P  =/=  A ) )
2725, 263anbi23d 1255 . . . . . . . . 9  |-  ( a  =  A  ->  (
( P  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  P  =/=  a
)  <->  ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
) ) )
28 opeq1 3796 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. a ,  x >.  =  <. A ,  x >. )
2928breq2d 4035 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( POutsideOf
<. a ,  x >.  <->  POutsideOf <. A ,  x >. ) )
3029rabbidv 2780 . . . . . . . . . 10  |-  ( a  =  A  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } )
3130eqeq2d 2294 . . . . . . . . 9  |-  ( a  =  A  ->  (
r  =  { x  e.  ( EE `  n
)  |  POutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } ) )
3227, 31anbi12d 691 . . . . . . . 8  |-  ( a  =  A  ->  (
( ( P  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  P  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } )  <->  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } ) ) )
3332rexbidv 2564 . . . . . . 7  |-  ( a  =  A  ->  ( E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  P  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } )  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } ) ) )
34 eqeq1 2289 . . . . . . . . 9  |-  ( r  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  ->  (
r  =  { x  e.  ( EE `  n
)  |  POutsideOf <. A ,  x >. }  <->  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) )
3534anbi2d 684 . . . . . . . 8  |-  ( r  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  ->  (
( ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } )  <->  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) ) )
3635rexbidv 2564 . . . . . . 7  |-  ( r  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  ->  ( E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } )  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) ) )
3724, 33, 36eloprabg 5935 . . . . . 6  |-  ( ( P  e.  ( EE
`  N )  /\  A  e.  ( EE `  N )  /\  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }  e.  _V )  ->  ( <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) ) )
3816, 37mp3an3 1266 . . . . 5  |-  ( ( P  e.  ( EE
`  N )  /\  A  e.  ( EE `  N ) )  -> 
( <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) ) )
3913, 14, 38syl2anc 642 . . . 4  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  -> 
( <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) ) )
4012, 39mpbird 223 . . 3  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) } )
41 df-br 4024 . . . . 5  |-  ( <. P ,  A >.Ray { x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }  <->  <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e. Ray )
42 df-ray 24761 . . . . . 6  |- Ray  =  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }
4342eleq2i 2347 . . . . 5  |-  ( <. <. P ,  A >. ,  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } >.  e. Ray  <->  <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) } )
4441, 43bitri 240 . . . 4  |-  ( <. P ,  A >.Ray { x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }  <->  <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) } )
45 funray 24763 . . . . 5  |-  Fun Ray
46 funbrfv 5561 . . . . 5  |-  ( Fun Ray  ->  ( <. P ,  A >.Ray { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. }  ->  (Ray `  <. P ,  A >. )  =  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } ) )
4745, 46ax-mp 8 . . . 4  |-  ( <. P ,  A >.Ray { x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }  ->  (Ray `  <. P ,  A >. )  =  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } )
4844, 47sylbir 204 . . 3  |-  ( <. <. P ,  A >. ,  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } >.  e.  { <. <.
p ,  a >. ,  r >.  |  E. n  e.  NN  (
( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }  ->  (Ray
`  <. P ,  A >. )  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } )
4940, 48syl 15 . 2  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  -> 
(Ray `  <. P ,  A >. )  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } )
501, 49syl5eq 2327 1  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  -> 
( PRay A )  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   _Vcvv 2788   <.cop 3643   class class class wbr 4023   Fun wfun 5249   ` cfv 5255  (class class class)co 5858   {coprab 5859   NNcn 9746   EEcee 24516  OutsideOfcoutsideof 24742  Raycray 24758
This theorem is referenced by:  lineunray  24770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-z 10025  df-uz 10231  df-fz 10783  df-ee 24519  df-ray 24761
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