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Theorem fvray 26080
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 6087 . 2  |-  ( PRay A )  =  (Ray
`  <. P ,  A >. )
2 eqid 2438 . . . . 5  |-  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }
3 fveq2 5731 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
43eleq2d 2505 . . . . . . . 8  |-  ( n  =  N  ->  ( P  e.  ( EE `  n )  <->  P  e.  ( EE `  N ) ) )
53eleq2d 2505 . . . . . . . 8  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
64, 53anbi12d 1256 . . . . . . 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 2952 . . . . . . . . 9  |-  ( ( EE `  n )  =  ( EE `  N )  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } )
83, 7syl 16 . . . . . . . 8  |-  ( n  =  N  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } )
98eqeq2d 2449 . . . . . . 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 693 . . . . . 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 3054 . . . . 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 667 . . . 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 964 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  P  e.  ( EE `  N ) )
14 simpr2 965 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  A  e.  ( EE `  N ) )
15 fvex 5745 . . . . . . 7  |-  ( EE
`  N )  e. 
_V
1615rabex 4357 . . . . . 6  |-  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  e.  _V
17 eleq1 2498 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p  e.  ( EE
`  n )  <->  P  e.  ( EE `  n ) ) )
18 neeq1 2611 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p  =/=  a  <->  P  =/=  a ) )
1917, 183anbi13d 1257 . . . . . . . . 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 4218 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
pOutsideOf <. a ,  x >.  <-> 
POutsideOf <. a ,  x >. ) )
2120rabbidv 2950 . . . . . . . . . 10  |-  ( p  =  P  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. a ,  x >. } )
2221eqeq2d 2449 . . . . . . . . 9  |-  ( p  =  P  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } ) )
2319, 22anbi12d 693 . . . . . . . 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 2728 . . . . . . 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 2498 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
26 neeq2 2612 . . . . . . . . . 10  |-  ( a  =  A  ->  ( P  =/=  a  <->  P  =/=  A ) )
2725, 263anbi23d 1258 . . . . . . . . 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 3986 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. a ,  x >.  =  <. A ,  x >. )
2928breq2d 4227 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( POutsideOf
<. a ,  x >.  <->  POutsideOf <. A ,  x >. ) )
3029rabbidv 2950 . . . . . . . . . 10  |-  ( a  =  A  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } )
3130eqeq2d 2449 . . . . . . . . 9  |-  ( a  =  A  ->  (
r  =  { x  e.  ( EE `  n
)  |  POutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } ) )
3227, 31anbi12d 693 . . . . . . . 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 2728 . . . . . . 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 2444 . . . . . . . . 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 686 . . . . . . . 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 2728 . . . . . . 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 6164 . . . . . 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 1269 . . . . 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 644 . . . 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 225 . . 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 4216 . . . . 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 26077 . . . . . 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 2502 . . . . 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 242 . . . 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 26079 . . . . 5  |-  Fun Ray
46 funbrfv 5768 . . . . 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 5 . . . 4  |-  ( <. P ,  A >.Ray { x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }  ->  (Ray `  <. P ,  A >. )  =  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } )
4844, 47sylbir 206 . . 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 16 . 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 2482 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711   _Vcvv 2958   <.cop 3819   class class class wbr 4215   Fun wfun 5451   ` cfv 5457  (class class class)co 6084   {coprab 6085   NNcn 10005   EEcee 25832  OutsideOfcoutsideof 26058  Raycray 26074
This theorem is referenced by:  lineunray  26086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-z 10288  df-uz 10494  df-fz 11049  df-ee 25835  df-ray 26077
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