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Theorem funray 24763
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 2707 . . . . . 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 955 . . . . . . . . . . 11  |-  ( ( p  e.  ( EE
`  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a )  ->  p  e.  ( EE `  n
) )
3 simp1 955 . . . . . . . . . . 11  |-  ( ( p  e.  ( EE
`  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a )  ->  p  e.  ( EE `  m
) )
4 axdimuniq 24541 . . . . . . . . . . . . . . 15  |-  ( ( ( n  e.  NN  /\  p  e.  ( EE
`  n ) )  /\  ( m  e.  NN  /\  p  e.  ( EE `  m
) ) )  ->  n  =  m )
5 fveq2 5525 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
6 rabeq 2782 . . . . . . . . . . . . . . . . . . 19  |-  ( ( EE `  n )  =  ( EE `  m )  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  m )  |  pOutsideOf <. a ,  x >. } )
75, 6syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  m  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  m )  |  pOutsideOf <. a ,  x >. } )
87eqeq2d 2294 . . . . . . . . . . . . . . . . 17  |-  ( n  =  m  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )
98anbi1d 685 . . . . . . . . . . . . . . . 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 2302 . . . . . . . . . . . . . . . 16  |-  ( ( r  =  { x  e.  ( EE `  m
)  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  r  =  s )
119, 10syl6bi 219 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  (
( r  =  {
x  e.  ( EE
`  n )  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  r  =  s ) )
124, 11syl 15 . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . 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 423 . . . . . . . . . . . 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 75 . . . . . . . . . . 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 463 . . . . . . . . . 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 418 . . . . . . . . 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 799 . . . . . . . 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 27 . . . . . . 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 2672 . . . . . 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 204 . . . . 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 1534 . . . 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 2289 . . . . . . . 8  |-  ( r  =  s  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  s  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) )
2423anbi2d 684 . . . . . . 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 2564 . . . . . 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 2350 . . . . . . . . 9  |-  ( n  =  m  ->  (
p  e.  ( EE
`  n )  <->  p  e.  ( EE `  m ) ) )
275eleq2d 2350 . . . . . . . . 9  |-  ( n  =  m  ->  (
a  e.  ( EE
`  n )  <->  a  e.  ( EE `  m ) ) )
2826, 273anbi12d 1253 . . . . . . . 8  |-  ( n  =  m  ->  (
( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  <->  ( p  e.  ( EE `  m
)  /\  a  e.  ( EE `  m )  /\  p  =/=  a
) ) )
297eqeq2d 2294 . . . . . . . 8  |-  ( n  =  m  ->  (
s  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )
3028, 29anbi12d 691 . . . . . . 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 2765 . . . . . 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 252 . . . . 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 2176 . . . 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 200 . . 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 5944 . 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 24761 . . 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 5275 . 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 200 1  |-  Fun Ray
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   E*wmo 2144    =/= wne 2446   E.wrex 2544   {crab 2547   <.cop 3643   class class class wbr 4023   Fun wfun 5249   ` cfv 5255   {coprab 5859   NNcn 9746   EEcee 24516  OutsideOfcoutsideof 24742  Raycray 24758
This theorem is referenced by:  fvray  24764
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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