Theorem rntpos 6236

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 2733	. . . . 5 |- x e. _V
2	1	elrn 4854	. . . 4 |- ( x e. ran tpos F <-> E. y ytpos F x )
3		vex 2733	. . . . . . . . 9 |- y e. _V
4	3, 1	breldm 4815	. . . . . . . 8 |- ( ytpos F x -> y e. dom tpos F )
5		dmtpos 6235	. . . . . . . . 9 |- ( Rel dom F -> dom tpos F = `' dom F )
6	5	eleq2d 2240	. . . . . . . 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 4989	. . . . . . . 8 |- Rel `' dom F
9		elrel 4713	. . . . . . . 8 |- ( ( Rel `' dom F /\ y e. `' dom F ) -> E. w E. z y = <. w , z >. )
10	8, 9	mpan 422	. . . . . . 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 3992	. . . . . . . . 9 |- ( y = <. w , z >. -> ( ytpos F x <-> <. w , z >.tpos F x ) )
13		vex 2733	. . . . . . . . . 10 |- w e. _V
14		vex 2733	. . . . . . . . . 10 |- z e. _V
15		brtposg 6233	. . . . . . . . . 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 1332	. . . . . . . . 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 4214	. . . . . . . . 9 |- <. z , w >. e. _V
19	18, 1	brelrn 4844	. . . . . . . 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 1889	. . . . . 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 1812	. . . 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 4854	. . . 4 |- ( x e. ran F <-> E. y y F x )
26	3, 1	breldm 4815	. . . . . . 7 |- ( y F x -> y e. dom F )
27		elrel 4713	. . . . . . . 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 3992	. . . . . . . . 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 4214	. . . . . . . . 9 |- <. w , z >. e. _V
33	32, 1	brelrn 4844	. . . . . . . 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 1889	. . . . . 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 1812	. . . 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 2168	1 |- ( Rel dom F -> ran tpos F = ran F )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   <.cop 3586   class class class wbr 3989   `'ccnv 4610   dom cdm 4611   ran crn 4612   Rel wrel 4616  tpos ctpos 6223

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-tpos 6224

This theorem is referenced by:  tposfo2  6246