Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  reltpos Structured version   Visualization version   GIF version

Theorem reltpos 7527
 Description: The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
reltpos Rel tpos 𝐹

Proof of Theorem reltpos
StepHypRef Expression
1 tposssxp 7526 . 2 tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
2 relxp 5283 . 2 Rel ((dom 𝐹 ∪ {∅}) × ran 𝐹)
3 relss 5363 . 2 (tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹) → (Rel ((dom 𝐹 ∪ {∅}) × ran 𝐹) → Rel tpos 𝐹))
41, 2, 3mp2 9 1 Rel tpos 𝐹
 Colors of variables: wff setvar class Syntax hints:   ∪ cun 3713   ⊆ wss 3715  ∅c0 4058  {csn 4321   × cxp 5264  ◡ccnv 5265  dom cdm 5266  ran crn 5267  Rel wrel 5271  tpos ctpos 7521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-tpos 7522 This theorem is referenced by:  brtpos2  7528  relbrtpos  7533  dftpos2  7539  dftpos3  7540  tpostpos  7542
 Copyright terms: Public domain W3C validator