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Theorem funray 24835
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 2720 . . . . . 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 24613 . . . . . . . . . . . . . . 15  |-  ( ( ( n  e.  NN  /\  p  e.  ( EE
`  n ) )  /\  ( m  e.  NN  /\  p  e.  ( EE `  m
) ) )  ->  n  =  m )
5 fveq2 5541 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
6 rabeq 2795 . . . . . . . . . . . . . . . . . . 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 2307 . . . . . . . . . . . . . . . . 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 2315 . . . . . . . . . . . . . . . 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 2685 . . . . . 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 1537 . . . 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 2302 . . . . . . . 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 2577 . . . . . 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 2363 . . . . . . . . 9  |-  ( n  =  m  ->  (
p  e.  ( EE
`  n )  <->  p  e.  ( EE `  m ) ) )
275eleq2d 2363 . . . . . . . . 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 2307 . . . . . . . 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 2778 . . . . . 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 2189 . . . 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 5960 . 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 24833 . . 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 5291 . 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 1530    = wceq 1632    e. wcel 1696   E*wmo 2157    =/= wne 2459   E.wrex 2557   {crab 2560   <.cop 3656   class class class wbr 4039   Fun wfun 5265   ` cfv 5271   {coprab 5875   NNcn 9762   EEcee 24588  OutsideOfcoutsideof 24814  Raycray 24830
This theorem is referenced by:  fvray  24836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-z 10041  df-uz 10247  df-fz 10799  df-ee 24591  df-ray 24833
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