Theorem rntpos 6194

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 2712	. . . . 5 |- x e. _V
2	1	elrn 4822	. . . 4 |- ( x e. ran tpos F <-> E. y ytpos F x )
3		vex 2712	. . . . . . . . 9 |- y e. _V
4	3, 1	breldm 4783	. . . . . . . 8 |- ( ytpos F x -> y e. dom tpos F )
5		dmtpos 6193	. . . . . . . . 9 |- ( Rel dom F -> dom tpos F = `' dom F )
6	5	eleq2d 2224	. . . . . . . 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 4957	. . . . . . . 8 |- Rel `' dom F
9		elrel 4681	. . . . . . . 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 3964	. . . . . . . . 9 |- ( y = <. w , z >. -> ( ytpos F x <-> <. w , z >.tpos F x ) )
13		vex 2712	. . . . . . . . . 10 |- w e. _V
14		vex 2712	. . . . . . . . . 10 |- z e. _V
15		brtposg 6191	. . . . . . . . . 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	bitrdi 195	. . . . . . . 8 |- ( y = <. w , z >. -> ( ytpos F x <-> <. z , w >. F x ) )
18	14, 13	opex 4184	. . . . . . . . 9 |- <. z , w >. e. _V
19	18, 1	brelrn 4812	. . . . . . . 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 1873	. . . . . 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 1796	. . . 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 4822	. . . 4 |- ( x e. ran F <-> E. y y F x )
26	3, 1	breldm 4783	. . . . . . 7 |- ( y F x -> y e. dom F )
27		elrel 4681	. . . . . . . 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 3964	. . . . . . . . 9 |- ( y = <. z , w >. -> ( y F x <-> <. z , w >. F x ) )
31	30, 16	bitr4di 197	. . . . . . . 8 |- ( y = <. z , w >. -> ( y F x <-> <. w , z >.tpos F x ) )
32	13, 14	opex 4184	. . . . . . . . 9 |- <. w , z >. e. _V
33	32, 1	brelrn 4812	. . . . . . . 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 1873	. . . . . 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 1796	. . . 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 2152	1 |- ( Rel dom F -> ran tpos F = ran F )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 2125   _Vcvv 2709   <.cop 3559   class class class wbr 3961   `'ccnv 4578   dom cdm 4579   ran crn 4580   Rel wrel 4584  tpos ctpos 6181

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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-fv 5171  df-tpos 6182

This theorem is referenced by:  tposfo2  6204