Theorem funline 32249

Description: Show that the Line relationship is a function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
funline	|- Fun Line

Proof of Theorem funline

Dummy variables a b k l m n 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 ( ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) /\ ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) <-> ( E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) /\ E. m e. NN ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) )
2		eqtr3 2643	. . . . . . . . 9 |- ( ( l = [ <. a , b >. ] `' Colinear /\ k = [ <. a , b >. ] `' Colinear ) -> l = k )
3	2	ad2ant2l 782	. . . . . . . 8 |- ( ( ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) /\ ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) -> l = k )
4	3	a1i 11	. . . . . . 7 |- ( ( n e. NN /\ m e. NN ) -> ( ( ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) /\ ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) -> l = k ) )
5	4	rexlimivv 3036	. . . . . 6 |- ( E. n e. NN E. m e. NN ( ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) /\ ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) -> l = k )
6	1, 5	sylbir 225	. . . . 5 |- ( ( E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) /\ E. m e. NN ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) -> l = k )
7	6	gen2 1723	. . . 4 |- A. l A. k ( ( E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) /\ E. m e. NN ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) -> l = k )
8		eqeq1 2626	. . . . . . . 8 |- ( l = k -> ( l = [ <. a , b >. ] `' Colinear <-> k = [ <. a , b >. ] `' Colinear ) )
9	8	anbi2d 740	. . . . . . 7 |- ( l = k -> ( ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) <-> ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) )
10	9	rexbidv 3052	. . . . . 6 |- ( l = k -> ( E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) <-> E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) )
11		fveq2 6191	. . . . . . . . . 10 |- ( n = m -> ( EE ` n ) = ( EE ` m ) )
12	11	eleq2d 2687	. . . . . . . . 9 |- ( n = m -> ( a e. ( EE ` n ) <-> a e. ( EE ` m ) ) )
13	11	eleq2d 2687	. . . . . . . . 9 |- ( n = m -> ( b e. ( EE ` n ) <-> b e. ( EE ` m ) ) )
14	12, 13	3anbi12d 1400	. . . . . . . 8 |- ( n = m -> ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) <-> ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) ) )
15	14	anbi1d 741	. . . . . . 7 |- ( n = m -> ( ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) <-> ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) )
16	15	cbvrexv 3172	. . . . . 6 |- ( E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) <-> E. m e. NN ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) )
17	10, 16	syl6bb 276	. . . . 5 |- ( l = k -> ( E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) <-> E. m e. NN ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) )
18	17	mo4 2517	. . . 4 |- ( E* l E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) <-> A. l A. k ( ( E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) /\ E. m e. NN ( ( a e. ( EE ` m ) /\ b e. ( EE ` m ) /\ a =/= b ) /\ k = [ <. a , b >. ] `' Colinear ) ) -> l = k ) )
19	7, 18	mpbir 221	. . 3 |- E* l E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear )
20	19	funoprab 6760	. 2 |- Fun { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) }
21		df-line2 32244	. . 3 |- Line = { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) }
22	21	funeqi 5909	. 2 |- ( Fun Line <-> Fun { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) } )
23	20, 22	mpbir 221	1 |- Fun Line

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   =/= wne 2794   E.wrex 2913   <.cop 4183   `'ccnv 5113   Fun wfun 5882   ` cfv 5888   {coprab 6651   [cec 7740   NNcn 11020   EEcee 25768   Colinear ccolin 32144  Linecline2 32241

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-iota 5851  df-fun 5890  df-fv 5896  df-oprab 6654  df-line2 32244

This theorem is referenced by:  fvline  32251