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Theorem fvray 26068
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 6077 . 2  |-  ( PRay A )  =  (Ray
`  <. P ,  A >. )
2 eqid 2436 . . . . 5  |-  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }
3 fveq2 5721 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
43eleq2d 2503 . . . . . . . 8  |-  ( n  =  N  ->  ( P  e.  ( EE `  n )  <->  P  e.  ( EE `  N ) ) )
53eleq2d 2503 . . . . . . . 8  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
64, 53anbi12d 1255 . . . . . . 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 2943 . . . . . . . . 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 2447 . . . . . . 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 692 . . . . . 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 3045 . . . . 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 666 . . . 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 963 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  P  e.  ( EE `  N ) )
14 simpr2 964 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  A  e.  ( EE `  N ) )
15 fvex 5735 . . . . . . 7  |-  ( EE
`  N )  e. 
_V
1615rabex 4347 . . . . . 6  |-  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  e.  _V
17 eleq1 2496 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p  e.  ( EE
`  n )  <->  P  e.  ( EE `  n ) ) )
18 neeq1 2607 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p  =/=  a  <->  P  =/=  a ) )
1917, 183anbi13d 1256 . . . . . . . . 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 4208 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
pOutsideOf <. a ,  x >.  <-> 
POutsideOf <. a ,  x >. ) )
2120rabbidv 2941 . . . . . . . . . 10  |-  ( p  =  P  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. a ,  x >. } )
2221eqeq2d 2447 . . . . . . . . 9  |-  ( p  =  P  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } ) )
2319, 22anbi12d 692 . . . . . . . 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 2719 . . . . . . 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 2496 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
26 neeq2 2608 . . . . . . . . . 10  |-  ( a  =  A  ->  ( P  =/=  a  <->  P  =/=  A ) )
2725, 263anbi23d 1257 . . . . . . . . 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 3977 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. a ,  x >.  =  <. A ,  x >. )
2928breq2d 4217 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( POutsideOf
<. a ,  x >.  <->  POutsideOf <. A ,  x >. ) )
3029rabbidv 2941 . . . . . . . . . 10  |-  ( a  =  A  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } )
3130eqeq2d 2447 . . . . . . . . 9  |-  ( a  =  A  ->  (
r  =  { x  e.  ( EE `  n
)  |  POutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } ) )
3227, 31anbi12d 692 . . . . . . . 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 2719 . . . . . . 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 2442 . . . . . . . . 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 685 . . . . . . . 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 2719 . . . . . . 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 6154 . . . . . 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 1268 . . . . 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 643 . . . 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 224 . . 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 4206 . . . . 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 26065 . . . . . 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 2500 . . . . 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 241 . . . 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 26067 . . . . 5  |-  Fun Ray
46 funbrfv 5758 . . . . 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 205 . . 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 2480 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2699   {crab 2702   _Vcvv 2949   <.cop 3810   class class class wbr 4205   Fun wfun 5441   ` cfv 5447  (class class class)co 6074   {coprab 6075   NNcn 9993   EEcee 25820  OutsideOfcoutsideof 26046  Raycray 26062
This theorem is referenced by:  lineunray  26074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-map 7013  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-z 10276  df-uz 10482  df-fz 11037  df-ee 25823  df-ray 26065
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