Theorem rntpos 6162

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 2692	. . . . 5 |- x e. _V
2	1	elrn 4790	. . . 4 |- ( x e. ran tpos F <-> E. y ytpos F x )
3		vex 2692	. . . . . . . . 9 |- y e. _V
4	3, 1	breldm 4751	. . . . . . . 8 |- ( ytpos F x -> y e. dom tpos F )
5		dmtpos 6161	. . . . . . . . 9 |- ( Rel dom F -> dom tpos F = `' dom F )
6	5	eleq2d 2210	. . . . . . . 8 |- ( Rel dom F -> ( y e. dom tpos F <-> y e. `' dom F ) )
7	4, 6	syl5ib 153	. . . . . . 7 |- ( Rel dom F -> ( ytpos F x -> y e. `' dom F ) )
8		relcnv 4925	. . . . . . . 8 |- Rel `' dom F
9		elrel 4649	. . . . . . . 8 |- ( ( Rel `' dom F /\ y e. `' dom F ) -> E. w E. z y = <. w , z >. )
10	8, 9	mpan 421	. . . . . . 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 3940	. . . . . . . . 9 |- ( y = <. w , z >. -> ( ytpos F x <-> <. w , z >.tpos F x ) )
13		vex 2692	. . . . . . . . . 10 |- w e. _V
14		vex 2692	. . . . . . . . . 10 |- z e. _V
15		brtposg 6159	. . . . . . . . . 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 1316	. . . . . . . . 9 |- ( <. w , z >.tpos F x <-> <. z , w >. F x )
17	12, 16	syl6bb 195	. . . . . . . 8 |- ( y = <. w , z >. -> ( ytpos F x <-> <. z , w >. F x ) )
18	14, 13	opex 4159	. . . . . . . . 9 |- <. z , w >. e. _V
19	18, 1	brelrn 4780	. . . . . . . 8 |- ( <. z , w >. F x -> x e. ran F )
20	17, 19	syl6bi 162	. . . . . . 7 |- ( y = <. w , z >. -> ( ytpos F x -> x e. ran F ) )
21	20	exlimivv 1869	. . . . . 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 1792	. . . 4 |- ( Rel dom F -> ( E. y ytpos F x -> x e. ran F ) )
24	2, 23	syl5bi 151	. . 3 |- ( Rel dom F -> ( x e. ran tpos F -> x e. ran F ) )
25	1	elrn 4790	. . . 4 |- ( x e. ran F <-> E. y y F x )
26	3, 1	breldm 4751	. . . . . . 7 |- ( y F x -> y e. dom F )
27		elrel 4649	. . . . . . . 8 |- ( ( Rel dom F /\ y e. dom F ) -> E. z E. w y = <. z , w >. )
28	27	ex 114	. . . . . . 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 3940	. . . . . . . . 9 |- ( y = <. z , w >. -> ( y F x <-> <. z , w >. F x ) )
31	30, 16	syl6bbr 197	. . . . . . . 8 |- ( y = <. z , w >. -> ( y F x <-> <. w , z >.tpos F x ) )
32	13, 14	opex 4159	. . . . . . . . 9 |- <. w , z >. e. _V
33	32, 1	brelrn 4780	. . . . . . . 8 |- ( <. w , z >.tpos F x -> x e. ran tpos F )
34	31, 33	syl6bi 162	. . . . . . 7 |- ( y = <. z , w >. -> ( y F x -> x e. ran tpos F ) )
35	34	exlimivv 1869	. . . . . 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 1792	. . . 4 |- ( Rel dom F -> ( E. y y F x -> x e. ran tpos F ) )
38	25, 37	syl5bi 151	. . 3 |- ( Rel dom F -> ( x e. ran F -> x e. ran tpos F ) )
39	24, 38	impbid 128	. 2 |- ( Rel dom F -> ( x e. ran tpos F <-> x e. ran F ) )
40	39	eqrdv 2138	1 |- ( Rel dom F -> ran tpos F = ran F )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   <.cop 3535   class class class wbr 3937   `'ccnv 4546   dom cdm 4547   ran crn 4548   Rel wrel 4552  tpos ctpos 6149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139  df-tpos 6150

This theorem is referenced by:  tposfo2  6172