Theorem rntpos 5813

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 2554	. . . . 5 |- x e. _V
2	1	elrn 4520	. . . 4 |- (x e. ran tpos F <-> E.y ytpos Fx)
3		vex 2554	. . . . . . . . 9 |- y e. _V
4	3, 1	breldm 4482	. . . . . . . 8 |- (ytpos Fx -> y e. dom tpos F)
5		dmtpos 5812	. . . . . . . . 9 |- ( Rel dom F -> dom tpos F = `' dom F)
6	5	eleq2d 2104	. . . . . . . 8 |- ( Rel dom F -> (y e. dom tpos F <-> y e. `' dom F))
7	4, 6	syl5ib 143	. . . . . . 7 |- ( Rel dom F -> (ytpos Fx -> y e. `' dom F))
8		relcnv 4646	. . . . . . . 8 |- Rel `' dom F
9		elrel 4385	. . . . . . . 8 |- (( Rel `' dom F /\ y e. `' dom F) -> E.wE.z y = <.w , z >.)
10	8, 9	mpan 400	. . . . . . 7 |- (y e. `' dom F -> E.wE.z y = <.w , z >.)
11	7, 10	syl6 29	. . . . . 6 |- ( Rel dom F -> (ytpos Fx -> E.wE.z y = <.w , z >.))
12		breq1 3758	. . . . . . . . 9 |- (y = <.w , z >. -> (ytpos Fx <-> <.w , z >.tpos Fx))
13		vex 2554	. . . . . . . . . 10 |- w e. _V
14		vex 2554	. . . . . . . . . 10 |- z e. _V
15		brtposg 5810	. . . . . . . . . 10 |- ((w e. _V /\ z e. _V /\ x e. _V) -> ( <.w , z >.tpos Fx <-> <.z , w >. Fx))
16	13, 14, 1, 15	mp3an 1231	. . . . . . . . 9 |- ( <.w , z >.tpos Fx <-> <.z , w >. Fx)
17	12, 16	syl6bb 185	. . . . . . . 8 |- (y = <.w , z >. -> (ytpos Fx <-> <.z , w >. Fx))
18	14, 13	opex 3957	. . . . . . . . 9 |- <.z , w >. e. _V
19	18, 1	brelrn 4510	. . . . . . . 8 |- ( <.z , w >. Fx -> x e. ran F)
20	17, 19	syl6bi 152	. . . . . . 7 |- (y = <.w , z >. -> (ytpos Fx -> x e. ran F))
21	20	exlimivv 1773	. . . . . 6 |- (E.wE.z y = <.w , z >. -> (ytpos Fx -> x e. ran F))
22	11, 21	syli 33	. . . . 5 |- ( Rel dom F -> (ytpos Fx -> x e. ran F))
23	22	exlimdv 1697	. . . 4 |- ( Rel dom F -> (E.y ytpos Fx -> 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 4520	. . . 4 |- (x e. ran F <-> E.y y Fx)
26	3, 1	breldm 4482	. . . . . . 7 |- (y Fx -> y e. dom F)
27		elrel 4385	. . . . . . . 8 |- (( Rel dom F /\ y e. dom F) -> E.zE.w y = <.z , w >.)
28	27	ex 108	. . . . . . 7 |- ( Rel dom F -> (y e. dom F -> E.zE.w y = <.z , w >.))
29	26, 28	syl5 28	. . . . . 6 |- ( Rel dom F -> (y Fx -> E.zE.w y = <.z , w >.))
30		breq1 3758	. . . . . . . . 9 |- (y = <.z , w >. -> (y Fx <-> <.z , w >. Fx))
31	30, 16	syl6bbr 187	. . . . . . . 8 |- (y = <.z , w >. -> (y Fx <-> <.w , z >.tpos Fx))
32	13, 14	opex 3957	. . . . . . . . 9 |- <.w , z >. e. _V
33	32, 1	brelrn 4510	. . . . . . . 8 |- ( <.w , z >.tpos Fx -> x e. ran tpos F)
34	31, 33	syl6bi 152	. . . . . . 7 |- (y = <.z , w >. -> (y Fx -> x e. ran tpos F))
35	34	exlimivv 1773	. . . . . 6 |- (E.zE.w y = <.z , w >. -> (y Fx -> x e. ran tpos F))
36	29, 35	syli 33	. . . . 5 |- ( Rel dom F -> (y Fx -> x e. ran tpos F))
37	36	exlimdv 1697	. . . 4 |- ( Rel dom F -> (E.y y Fx -> 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 2035	1 |- ( Rel dom F -> ran tpos F = ran F)

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1242  E.wex 1378   e. wcel 1390   _Vcvv 2551   <.cop 3370   class class class wbr 3755   `'ccnv 4287   dom cdm 4288   ran crn 4289   Rel wrel 4293  tpos ctpos 5800

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-tpos 5801

This theorem is referenced by:  tposfo2  5823