Theorem rntpos 7365

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 3203	. . . . 5 |- z e. _V
2	1	elrn 5366	. . . 4 |- ( z e. ran tpos F <-> E. w wtpos F z )
3		vex 3203	. . . . . . . . 9 |- w e. _V
4	3, 1	breldm 5329	. . . . . . . 8 |- ( wtpos F z -> w e. dom tpos F )
5		dmtpos 7364	. . . . . . . . 9 |- ( Rel dom F -> dom tpos F = `' dom F )
6	5	eleq2d 2687	. . . . . . . 8 |- ( Rel dom F -> ( w e. dom tpos F <-> w e. `' dom F ) )
7	4, 6	syl5ib 234	. . . . . . 7 |- ( Rel dom F -> ( wtpos F z -> w e. `' dom F ) )
8		relcnv 5503	. . . . . . . 8 |- Rel `' dom F
9		elrel 5222	. . . . . . . 8 |- ( ( Rel `' dom F /\ w e. `' dom F ) -> E. x E. y w = <. x , y >. )
10	8, 9	mpan 706	. . . . . . 7 |- ( w e. `' dom F -> E. x E. y w = <. x , y >. )
11	7, 10	syl6 35	. . . . . 6 |- ( Rel dom F -> ( wtpos F z -> E. x E. y w = <. x , y >. ) )
12		breq1 4656	. . . . . . . . 9 |- ( w = <. x , y >. -> ( wtpos F z <-> <. x , y >.tpos F z ) )
13		brtpos 7361	. . . . . . . . . 10 |- ( z e. _V -> ( <. x , y >.tpos F z <-> <. y , x >. F z ) )
14	1, 13	ax-mp 5	. . . . . . . . 9 |- ( <. x , y >.tpos F z <-> <. y , x >. F z )
15	12, 14	syl6bb 276	. . . . . . . 8 |- ( w = <. x , y >. -> ( wtpos F z <-> <. y , x >. F z ) )
16		opex 4932	. . . . . . . . 9 |- <. y , x >. e. _V
17	16, 1	brelrn 5356	. . . . . . . 8 |- ( <. y , x >. F z -> z e. ran F )
18	15, 17	syl6bi 243	. . . . . . 7 |- ( w = <. x , y >. -> ( wtpos F z -> z e. ran F ) )
19	18	exlimivv 1860	. . . . . 6 |- ( E. x E. y w = <. x , y >. -> ( wtpos F z -> z e. ran F ) )
20	11, 19	syli 39	. . . . 5 |- ( Rel dom F -> ( wtpos F z -> z e. ran F ) )
21	20	exlimdv 1861	. . . 4 |- ( Rel dom F -> ( E. w wtpos F z -> z e. ran F ) )
22	2, 21	syl5bi 232	. . 3 |- ( Rel dom F -> ( z e. ran tpos F -> z e. ran F ) )
23	1	elrn 5366	. . . 4 |- ( z e. ran F <-> E. w w F z )
24	3, 1	breldm 5329	. . . . . . 7 |- ( w F z -> w e. dom F )
25		elrel 5222	. . . . . . . 8 |- ( ( Rel dom F /\ w e. dom F ) -> E. y E. x w = <. y , x >. )
26	25	ex 450	. . . . . . 7 |- ( Rel dom F -> ( w e. dom F -> E. y E. x w = <. y , x >. ) )
27	24, 26	syl5 34	. . . . . 6 |- ( Rel dom F -> ( w F z -> E. y E. x w = <. y , x >. ) )
28		breq1 4656	. . . . . . . . 9 |- ( w = <. y , x >. -> ( w F z <-> <. y , x >. F z ) )
29	28, 14	syl6bbr 278	. . . . . . . 8 |- ( w = <. y , x >. -> ( w F z <-> <. x , y >.tpos F z ) )
30		opex 4932	. . . . . . . . 9 |- <. x , y >. e. _V
31	30, 1	brelrn 5356	. . . . . . . 8 |- ( <. x , y >.tpos F z -> z e. ran tpos F )
32	29, 31	syl6bi 243	. . . . . . 7 |- ( w = <. y , x >. -> ( w F z -> z e. ran tpos F ) )
33	32	exlimivv 1860	. . . . . 6 |- ( E. y E. x w = <. y , x >. -> ( w F z -> z e. ran tpos F ) )
34	27, 33	syli 39	. . . . 5 |- ( Rel dom F -> ( w F z -> z e. ran tpos F ) )
35	34	exlimdv 1861	. . . 4 |- ( Rel dom F -> ( E. w w F z -> z e. ran tpos F ) )
36	23, 35	syl5bi 232	. . 3 |- ( Rel dom F -> ( z e. ran F -> z e. ran tpos F ) )
37	22, 36	impbid 202	. 2 |- ( Rel dom F -> ( z e. ran tpos F <-> z e. ran F ) )
38	37	eqrdv 2620	1 |- ( Rel dom F -> ran tpos F = ran F )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   Rel wrel 5119  tpos ctpos 7351

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

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-tpos 7352

This theorem is referenced by:  tposfo2  7375  oppchofcl  16900  oyoncl  16910