Theorem funray 32247

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.

Step	Hyp	Ref	Expression
1		reeanv 3107	. . . . . 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 1061	. . . . . . . . . . 11 |- ( ( p e. ( EE ` n ) /\ a e. ( EE ` n ) /\ p =/= a ) -> p e. ( EE ` n ) )
3		simp1 1061	. . . . . . . . . . 11 |- ( ( p e. ( EE ` m ) /\ a e. ( EE ` m ) /\ p =/= a ) -> p e. ( EE ` m ) )
4		axdimuniq 25793	. . . . . . . . . . . . . . 15 |- ( ( ( n e. NN /\ p e. ( EE ` n ) ) /\ ( m e. NN /\ p e. ( EE ` m ) ) ) -> n = m )
5		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 |- ( n = m -> ( EE ` n ) = ( EE ` m ) )
6		rabeq 3192	. . . . . . . . . . . . . . . . . . 19 |- ( ( EE ` n ) = ( EE ` m ) -> { x e. ( EE ` n ) | pOutsideOf <. a , x >. } = { x e. ( EE ` m ) | pOutsideOf <. a , x >. } )
7	5, 6	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( n = m -> { x e. ( EE ` n ) | pOutsideOf <. a , x >. } = { x e. ( EE ` m ) | pOutsideOf <. a , x >. } )
8	7	eqeq2d 2632	. . . . . . . . . . . . . . . . 17 |- ( n = m -> ( r = { x e. ( EE ` n ) | pOutsideOf <. a , x >. } <-> r = { x e. ( EE ` m ) | pOutsideOf <. a , x >. } ) )
9	8	anbi1d 741	. . . . . . . . . . . . . . . 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 2643	. . . . . . . . . . . . . . . 16 |- ( ( r = { x e. ( EE ` m ) | pOutsideOf <. a , x >. } /\ s = { x e. ( EE ` m ) | pOutsideOf <. a , x >. } ) -> r = s )
11	9, 10	syl6bi 243	. . . . . . . . . . . . . . 15 |- ( n = m -> ( ( r = { x e. ( EE ` n ) | pOutsideOf <. a , x >. } /\ s = { x e. ( EE ` m ) | pOutsideOf <. a , x >. } ) -> r = s ) )
12	4, 11	syl 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 ) )
13	12	an4s 869	. . . . . . . . . . . . 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 ) )
14	13	ex 450	. . . . . . . . . . . 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 ) ) )
15	14	com3l 89	. . . . . . . . . . 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 ) ) )
16	2, 3, 15	syl2an 494	. . . . . . . . . 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 ) ) )
17	16	imp 445	. . . . . . . . 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 ) )
18	17	an4s 869	. . . . . . . 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 ) )
19	18	com12 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 ) )
20	19	rexlimivv 3036	. . . . . 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 )
21	1, 20	sylbir 225	. . . . 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 )
22	21	gen2 1723	. . . 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 2626	. . . . . . . 8 |- ( r = s -> ( r = { x e. ( EE ` n ) | pOutsideOf <. a , x >. } <-> s = { x e. ( EE ` n ) | pOutsideOf <. a , x >. } ) )
24	23	anbi2d 740	. . . . . . 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 >. } ) ) )
25	24	rexbidv 3052	. . . . . 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 >. } ) ) )
26	5	eleq2d 2687	. . . . . . . . 9 |- ( n = m -> ( p e. ( EE ` n ) <-> p e. ( EE ` m ) ) )
27	5	eleq2d 2687	. . . . . . . . 9 |- ( n = m -> ( a e. ( EE ` n ) <-> a e. ( EE ` m ) ) )
28	26, 27	3anbi12d 1400	. . . . . . . 8 |- ( n = m -> ( ( p e. ( EE ` n ) /\ a e. ( EE ` n ) /\ p =/= a ) <-> ( p e. ( EE ` m ) /\ a e. ( EE ` m ) /\ p =/= a ) ) )
29	7	eqeq2d 2632	. . . . . . . 8 |- ( n = m -> ( s = { x e. ( EE ` n ) | pOutsideOf <. a , x >. } <-> s = { x e. ( EE ` m ) | pOutsideOf <. a , x >. } ) )
30	28, 29	anbi12d 747	. . . . . . 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 >. } ) ) )
31	30	cbvrexv 3172	. . . . . 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 >. } ) )
32	25, 31	syl6bb 276	. . . . 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 >. } ) ) )
33	32	mo4 2517	. . . 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 ) )
34	22, 33	mpbir 221	. . 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 >. } )
35	34	funoprab 6760	. 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 32245	. . 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 >. } ) }
37	36	funeqi 5909	. 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 >. } ) } )
38	35, 37	mpbir 221	1 |- Fun Ray

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   =/= wne 2794   E.wrex 2913   {crab 2916   <.cop 4183   class class class wbr 4653   Fun wfun 5882   ` cfv 5888   {coprab 6651   NNcn 11020   EEcee 25768  OutsideOfcoutsideof 32226  Raycray 32242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-z 11378  df-uz 11688  df-fz 12327  df-ee 25771  df-ray 32245

This theorem is referenced by:  fvray  32248