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Theorem funray 30899
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 2996 . . . . . 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 1005 . . . . . . . . . . 11  |-  ( ( p  e.  ( EE
`  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a )  ->  p  e.  ( EE `  n
) )
3 simp1 1005 . . . . . . . . . . 11  |-  ( ( p  e.  ( EE
`  m )  /\  a  e.  ( EE `  m )  /\  p  =/=  a )  ->  p  e.  ( EE `  m
) )
4 axdimuniq 24929 . . . . . . . . . . . . . . 15  |-  ( ( ( n  e.  NN  /\  p  e.  ( EE
`  n ) )  /\  ( m  e.  NN  /\  p  e.  ( EE `  m
) ) )  ->  n  =  m )
5 fveq2 5877 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
6 rabeq 3074 . . . . . . . . . . . . . . . . . . 19  |-  ( ( EE `  n )  =  ( EE `  m )  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  m )  |  pOutsideOf <. a ,  x >. } )
75, 6syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  m  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  m )  |  pOutsideOf <. a ,  x >. } )
87eqeq2d 2436 . . . . . . . . . . . . . . . . 17  |-  ( n  =  m  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )
98anbi1d 709 . . . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . . 16  |-  ( ( r  =  { x  e.  ( EE `  m
)  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  r  =  s )
119, 10syl6bi 231 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  (
( r  =  {
x  e.  ( EE
`  n )  |  pOutsideOf <. a ,  x >. }  /\  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } )  ->  r  =  s ) )
124, 11syl 17 . . . . . . . . . . . . . 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 833 . . . . . . . . . . . . 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 435 . . . . . . . . . . . 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 84 . . . . . . . . . . 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 479 . . . . . . . . . 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 430 . . . . . . . . 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 833 . . . . . . . 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 32 . . . . . . 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 2922 . . . . . 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 216 . . . . 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 1666 . . . 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 2426 . . . . . . . 8  |-  ( r  =  s  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  s  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) )
2423anbi2d 708 . . . . . . 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 2939 . . . . . 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 2492 . . . . . . . . 9  |-  ( n  =  m  ->  (
p  e.  ( EE
`  n )  <->  p  e.  ( EE `  m ) ) )
275eleq2d 2492 . . . . . . . . 9  |-  ( n  =  m  ->  (
a  e.  ( EE
`  n )  <->  a  e.  ( EE `  m ) ) )
2826, 273anbi12d 1336 . . . . . . . 8  |-  ( n  =  m  ->  (
( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  <->  ( p  e.  ( EE `  m
)  /\  a  e.  ( EE `  m )  /\  p  =/=  a
) ) )
297eqeq2d 2436 . . . . . . . 8  |-  ( n  =  m  ->  (
s  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  s  =  { x  e.  ( EE `  m )  |  pOutsideOf <. a ,  x >. } ) )
3028, 29anbi12d 715 . . . . . . 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 3056 . . . . . 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 264 . . . . 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 2313 . . . 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 212 . . 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 6406 . 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 30897 . . 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 5617 . 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 212 1  |-  Fun Ray
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1868   E*wmo 2266    =/= wne 2618   E.wrex 2776   {crab 2779   <.cop 4002   class class class wbr 4420   Fun wfun 5591   ` cfv 5597   {coprab 6302   NNcn 10609   EEcee 24904  OutsideOfcoutsideof 30878  Raycray 30894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-map 7478  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-z 10938  df-uz 11160  df-fz 11785  df-ee 24907  df-ray 30897
This theorem is referenced by:  fvray  30900
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