Theorem rntpos 6022

Description: The range of tpos F when dom F is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
rntpos	|- ( Rel dom F -> ran tpos F = ran F )

Proof of Theorem rntpos

Dummy variables x y w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2622	. . . . 5 |- x e. _V
2	1	elrn 4678	. . . 4 |- ( x e. ran tpos F <-> E. y ytpos F x )
3		vex 2622	. . . . . . . . 9 |- y e. _V
4	3, 1	breldm 4640	. . . . . . . 8 |- ( ytpos F x -> y e. dom tpos F )
5		dmtpos 6021	. . . . . . . . 9 |- ( Rel dom F -> dom tpos F = `' dom F )
6	5	eleq2d 2157	. . . . . . . 8 |- ( Rel dom F -> ( y e. dom tpos F <-> y e. `' dom F ) )
7	4, 6	syl5ib 152	. . . . . . 7 |- ( Rel dom F -> ( ytpos F x -> y e. `' dom F ) )
8		relcnv 4810	. . . . . . . 8 |- Rel `' dom F
9		elrel 4540	. . . . . . . 8 |- ( ( Rel `' dom F /\ y e. `' dom F ) -> E. w E. z y = <. w , z >. )
10	8, 9	mpan 415	. . . . . . 7 |- ( y e. `' dom F -> E. w E. z y = <. w , z >. )
11	7, 10	syl6 33	. . . . . 6 |- ( Rel dom F -> ( ytpos F x -> E. w E. z y = <. w , z >. ) )
12		breq1 3848	. . . . . . . . 9 |- ( y = <. w , z >. -> ( ytpos F x <-> <. w , z >.tpos F x ) )
13		vex 2622	. . . . . . . . . 10 |- w e. _V
14		vex 2622	. . . . . . . . . 10 |- z e. _V
15		brtposg 6019	. . . . . . . . . 10 |- ( ( w e. _V /\ z e. _V /\ x e. _V ) -> ( <. w , z >.tpos F x <-> <. z , w >. F x ) )
16	13, 14, 1, 15	mp3an 1273	. . . . . . . . 9 |- ( <. w , z >.tpos F x <-> <. z , w >. F x )
17	12, 16	syl6bb 194	. . . . . . . 8 |- ( y = <. w , z >. -> ( ytpos F x <-> <. z , w >. F x ) )
18	14, 13	opex 4056	. . . . . . . . 9 |- <. z , w >. e. _V
19	18, 1	brelrn 4668	. . . . . . . 8 |- ( <. z , w >. F x -> x e. ran F )
20	17, 19	syl6bi 161	. . . . . . 7 |- ( y = <. w , z >. -> ( ytpos F x -> x e. ran F ) )
21	20	exlimivv 1824	. . . . . 6 |- ( E. w E. z y = <. w , z >. -> ( ytpos F x -> x e. ran F ) )
22	11, 21	syli 37	. . . . 5 |- ( Rel dom F -> ( ytpos F x -> x e. ran F ) )
23	22	exlimdv 1747	. . . 4 |- ( Rel dom F -> ( E. y ytpos F x -> x e. ran F ) )
24	2, 23	syl5bi 150	. . 3 |- ( Rel dom F -> ( x e. ran tpos F -> x e. ran F ) )
25	1	elrn 4678	. . . 4 |- ( x e. ran F <-> E. y y F x )
26	3, 1	breldm 4640	. . . . . . 7 |- ( y F x -> y e. dom F )
27		elrel 4540	. . . . . . . 8 |- ( ( Rel dom F /\ y e. dom F ) -> E. z E. w y = <. z , w >. )
28	27	ex 113	. . . . . . 7 |- ( Rel dom F -> ( y e. dom F -> E. z E. w y = <. z , w >. ) )
29	26, 28	syl5 32	. . . . . 6 |- ( Rel dom F -> ( y F x -> E. z E. w y = <. z , w >. ) )
30		breq1 3848	. . . . . . . . 9 |- ( y = <. z , w >. -> ( y F x <-> <. z , w >. F x ) )
31	30, 16	syl6bbr 196	. . . . . . . 8 |- ( y = <. z , w >. -> ( y F x <-> <. w , z >.tpos F x ) )
32	13, 14	opex 4056	. . . . . . . . 9 |- <. w , z >. e. _V
33	32, 1	brelrn 4668	. . . . . . . 8 |- ( <. w , z >.tpos F x -> x e. ran tpos F )
34	31, 33	syl6bi 161	. . . . . . 7 |- ( y = <. z , w >. -> ( y F x -> x e. ran tpos F ) )
35	34	exlimivv 1824	. . . . . 6 |- ( E. z E. w y = <. z , w >. -> ( y F x -> x e. ran tpos F ) )
36	29, 35	syli 37	. . . . 5 |- ( Rel dom F -> ( y F x -> x e. ran tpos F ) )
37	36	exlimdv 1747	. . . 4 |- ( Rel dom F -> ( E. y y F x -> x e. ran tpos F ) )
38	25, 37	syl5bi 150	. . 3 |- ( Rel dom F -> ( x e. ran F -> x e. ran tpos F ) )
39	24, 38	impbid 127	. 2 |- ( Rel dom F -> ( x e. ran tpos F <-> x e. ran F ) )
40	39	eqrdv 2086	1 |- ( Rel dom F -> ran tpos F = ran F )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   E.wex 1426   e. wcel 1438   _Vcvv 2619   <.cop 3449   class class class wbr 3845   `'ccnv 4437   dom cdm 4438   ran crn 4439   Rel wrel 4443  tpos ctpos 6009

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023  df-tpos 6010

This theorem is referenced by:  tposfo2  6032