Theorem rntpos 5872

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 2560	. . . . 5 |- x e. _V
2	1	elrn 4577	. . . 4 |- ( x e. ran tpos F <-> E. y ytpos F x )
3		vex 2560	. . . . . . . . 9 |- y e. _V
4	3, 1	breldm 4539	. . . . . . . 8 |- ( ytpos F x -> y e. dom tpos F )
5		dmtpos 5871	. . . . . . . . 9 |- ( Rel dom F -> dom tpos F = `' dom F )
6	5	eleq2d 2107	. . . . . . . 8 |- ( Rel dom F -> ( y e. dom tpos F <-> y e. `' dom F ) )
7	4, 6	syl5ib 143	. . . . . . 7 |- ( Rel dom F -> ( ytpos F x -> y e. `' dom F ) )
8		relcnv 4703	. . . . . . . 8 |- Rel `' dom F
9		elrel 4442	. . . . . . . 8 |- ( ( Rel `' dom F /\ y e. `' dom F ) -> E. w E. z y = <. w , z >. )
10	8, 9	mpan 400	. . . . . . 7 |- ( y e. `' dom F -> E. w E. z y = <. w , z >. )
11	7, 10	syl6 29	. . . . . 6 |- ( Rel dom F -> ( ytpos F x -> E. w E. z y = <. w , z >. ) )
12		breq1 3767	. . . . . . . . 9 |- ( y = <. w , z >. -> ( ytpos F x <-> <. w , z >.tpos F x ) )
13		vex 2560	. . . . . . . . . 10 |- w e. _V
14		vex 2560	. . . . . . . . . 10 |- z e. _V
15		brtposg 5869	. . . . . . . . . 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 1232	. . . . . . . . 9 |- ( <. w , z >.tpos F x <-> <. z , w >. F x )
17	12, 16	syl6bb 185	. . . . . . . 8 |- ( y = <. w , z >. -> ( ytpos F x <-> <. z , w >. F x ) )
18	14, 13	opex 3966	. . . . . . . . 9 |- <. z , w >. e. _V
19	18, 1	brelrn 4567	. . . . . . . 8 |- ( <. z , w >. F x -> x e. ran F )
20	17, 19	syl6bi 152	. . . . . . 7 |- ( y = <. w , z >. -> ( ytpos F x -> x e. ran F ) )
21	20	exlimivv 1776	. . . . . 6 |- ( E. w E. z y = <. w , z >. -> ( ytpos F x -> x e. ran F ) )
22	11, 21	syli 33	. . . . 5 |- ( Rel dom F -> ( ytpos F x -> x e. ran F ) )
23	22	exlimdv 1700	. . . 4 |- ( Rel dom F -> ( E. y ytpos F x -> x e. ran F ) )
24	2, 23	syl5bi 141	. . 3 |- ( Rel dom F -> ( x e. ran tpos F -> x e. ran F ) )
25	1	elrn 4577	. . . 4 |- ( x e. ran F <-> E. y y F x )
26	3, 1	breldm 4539	. . . . . . 7 |- ( y F x -> y e. dom F )
27		elrel 4442	. . . . . . . 8 |- ( ( Rel dom F /\ y e. dom F ) -> E. z E. w y = <. z , w >. )
28	27	ex 108	. . . . . . 7 |- ( Rel dom F -> ( y e. dom F -> E. z E. w y = <. z , w >. ) )
29	26, 28	syl5 28	. . . . . 6 |- ( Rel dom F -> ( y F x -> E. z E. w y = <. z , w >. ) )
30		breq1 3767	. . . . . . . . 9 |- ( y = <. z , w >. -> ( y F x <-> <. z , w >. F x ) )
31	30, 16	syl6bbr 187	. . . . . . . 8 |- ( y = <. z , w >. -> ( y F x <-> <. w , z >.tpos F x ) )
32	13, 14	opex 3966	. . . . . . . . 9 |- <. w , z >. e. _V
33	32, 1	brelrn 4567	. . . . . . . 8 |- ( <. w , z >.tpos F x -> x e. ran tpos F )
34	31, 33	syl6bi 152	. . . . . . 7 |- ( y = <. z , w >. -> ( y F x -> x e. ran tpos F ) )
35	34	exlimivv 1776	. . . . . 6 |- ( E. z E. w y = <. z , w >. -> ( y F x -> x e. ran tpos F ) )
36	29, 35	syli 33	. . . . 5 |- ( Rel dom F -> ( y F x -> x e. ran tpos F ) )
37	36	exlimdv 1700	. . . 4 |- ( Rel dom F -> ( E. y y F x -> x e. ran tpos F ) )
38	25, 37	syl5bi 141	. . 3 |- ( Rel dom F -> ( x e. ran F -> x e. ran tpos F ) )
39	24, 38	impbid 120	. 2 |- ( Rel dom F -> ( x e. ran tpos F <-> x e. ran F ) )
40	39	eqrdv 2038	1 |- ( Rel dom F -> ran tpos F = ran F )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557   <.cop 3378   class class class wbr 3764   `'ccnv 4344   dom cdm 4345   ran crn 4346   Rel wrel 4350  tpos ctpos 5859

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-tpos 5860

This theorem is referenced by:  tposfo2  5882