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Theorem funray 28169
Description: Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funray  |-  Fun Ray

Proof of Theorem funray
Dummy variables  m  a  n  p  r 
s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reeanv 2886 . . . . . 6  |-  ( E. n  e.  NN  E. m  e.  NN  (
( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  /\  (
( p  e.  ( EE `  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a
)  /\  s  =  { x  e.  ( EE `  m )  |  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. m  e.  NN  (
( p  e.  ( EE `  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a
)  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) ) )
2 simp1 988 . . . . . . . . . . 11  |-  ( ( p  e.  ( EE
`  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a )  ->  p  e.  ( EE `  n
) )
3 simp1 988 . . . . . . . . . . 11  |-  ( ( p  e.  ( EE
`  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a )  ->  p  e.  ( EE `  m
) )
4 axdimuniq 23157 . . . . . . . . . . . . . . 15  |-  ( ( ( n  e.  NN  /\  p  e.  ( EE
`  n ) )  /\  ( m  e.  NN  /\  p  e.  ( EE `  m
) ) )  ->  n  =  m )
5 fveq2 5689 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
6 rabeq 2964 . . . . . . . . . . . . . . . . . . 19  |-  ( ( EE `  n )  =  ( EE `  m )  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  m )  |  pOutsideOf <. a ,  x >. } )
75, 6syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  m  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  m )  |  pOutsideOf <. a ,  x >. } )
87eqeq2d 2452 . . . . . . . . . . . . . . . . 17  |-  ( n  =  m  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )
98anbi1d 704 . . . . . . . . . . . . . . . 16  |-  ( n  =  m  ->  (
( r  =  {
x  e.  ( EE
`  n )  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  <->  ( r  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) ) )
10 eqtr3 2460 . . . . . . . . . . . . . . . 16  |-  ( ( r  =  { x  e.  ( EE `  m
)  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  r  =  s )
119, 10syl6bi 228 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  (
( r  =  {
x  e.  ( EE
`  n )  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  r  =  s ) )
124, 11syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( n  e.  NN  /\  p  e.  ( EE
`  n ) )  /\  ( m  e.  NN  /\  p  e.  ( EE `  m
) ) )  -> 
( ( r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  r  =  s ) )
1312an4s 822 . . . . . . . . . . . . 13  |-  ( ( ( n  e.  NN  /\  m  e.  NN )  /\  ( p  e.  ( EE `  n
)  /\  p  e.  ( EE `  m ) ) )  ->  (
( r  =  {
x  e.  ( EE
`  n )  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  r  =  s ) )
1413ex 434 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  m  e.  NN )  ->  ( ( p  e.  ( EE `  n
)  /\  p  e.  ( EE `  m ) )  ->  ( (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  r  =  s ) ) )
1514com3l 81 . . . . . . . . . . 11  |-  ( ( p  e.  ( EE
`  n )  /\  p  e.  ( EE `  m ) )  -> 
( ( r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  (
( n  e.  NN  /\  m  e.  NN )  ->  r  =  s ) ) )
162, 3, 15syl2an 477 . . . . . . . . . 10  |-  ( ( ( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  ( p  e.  ( EE `  m
)  /\  a  e.  ( EE `  m )  /\  p  =/=  a
) )  ->  (
( r  =  {
x  e.  ( EE
`  n )  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  (
( n  e.  NN  /\  m  e.  NN )  ->  r  =  s ) ) )
1716imp 429 . . . . . . . . 9  |-  ( ( ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  ( p  e.  ( EE `  m
)  /\  a  e.  ( EE `  m )  /\  p  =/=  a
) )  /\  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )  -> 
( ( n  e.  NN  /\  m  e.  NN )  ->  r  =  s ) )
1817an4s 822 . . . . . . . 8  |-  ( ( ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  /\  (
( p  e.  ( EE `  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a
)  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )  -> 
( ( n  e.  NN  /\  m  e.  NN )  ->  r  =  s ) )
1918com12 31 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  NN )  ->  ( ( ( ( p  e.  ( EE
`  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a )  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  /\  (
( p  e.  ( EE `  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a
)  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )  -> 
r  =  s ) )
2019rexlimivv 2844 . . . . . 6  |-  ( E. n  e.  NN  E. m  e.  NN  (
( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  /\  (
( p  e.  ( EE `  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a
)  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )  -> 
r  =  s )
211, 20sylbir 213 . . . . 5  |-  ( ( E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  /\  E. m  e.  NN  (
( p  e.  ( EE `  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a
)  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )  -> 
r  =  s )
2221gen2 1592 . . . 4  |-  A. r A. s ( ( E. n  e.  NN  (
( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  /\  E. m  e.  NN  (
( p  e.  ( EE `  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a
)  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )  -> 
r  =  s )
23 eqeq1 2447 . . . . . . . 8  |-  ( r  =  s  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  s  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) )
2423anbi2d 703 . . . . . . 7  |-  ( r  =  s  ->  (
( ( 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 )  /\  s  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) ) )
2524rexbidv 2734 . . . . . 6  |-  ( r  =  s  ->  ( 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 )  /\  s  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) ) )
265eleq2d 2508 . . . . . . . . 9  |-  ( n  =  m  ->  (
p  e.  ( EE
`  n )  <->  p  e.  ( EE `  m ) ) )
275eleq2d 2508 . . . . . . . . 9  |-  ( n  =  m  ->  (
a  e.  ( EE
`  n )  <->  a  e.  ( EE `  m ) ) )
2826, 273anbi12d 1290 . . . . . . . 8  |-  ( n  =  m  ->  (
( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  <->  ( p  e.  ( EE `  m
)  /\  a  e.  ( EE `  m )  /\  p  =/=  a
) ) )
297eqeq2d 2452 . . . . . . . 8  |-  ( n  =  m  ->  (
s  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )
3028, 29anbi12d 710 . . . . . . 7  |-  ( n  =  m  ->  (
( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  s  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  <->  ( (
p  e.  ( EE
`  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a )  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) ) )
3130cbvrexv 2946 . . . . . 6  |-  ( E. n  e.  NN  (
( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  s  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  <->  E. m  e.  NN  ( ( p  e.  ( EE `  m )  /\  a  e.  ( EE `  m
)  /\  p  =/=  a )  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )
3225, 31syl6bb 261 . . . . 5  |-  ( r  =  s  ->  ( E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  <->  E. m  e.  NN  ( ( p  e.  ( EE `  m )  /\  a  e.  ( EE `  m
)  /\  p  =/=  a )  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) ) )
3332mo4 2317 . . . 4  |-  ( E* r E. n  e.  NN  ( ( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n
)  /\  p  =/=  a )  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  <->  A. r A. s ( ( E. n  e.  NN  (
( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  /\  E. m  e.  NN  (
( p  e.  ( EE `  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a
)  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )  -> 
r  =  s ) )
3422, 33mpbir 209 . . 3  |-  E* r E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )
3534funoprab 6188 . 2  |-  Fun  { <. <. 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 >. } ) }
36 df-ray 28167 . . 3  |- 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 >. } ) }
3736funeqi 5436 . 2  |-  ( Fun Ray  <->  Fun 
{ <. <. 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 >. } ) } )
3835, 37mpbir 209 1  |-  Fun Ray
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756   E*wmo 2254    =/= wne 2604   E.wrex 2714   {crab 2717   <.cop 3881   class class class wbr 4290   Fun wfun 5410   ` cfv 5416   {coprab 6090   NNcn 10320   EEcee 23132  OutsideOfcoutsideof 28148  Raycray 28164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-z 10645  df-uz 10860  df-fz 11436  df-ee 23135  df-ray 28167
This theorem is referenced by:  fvray  28170
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