Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfcoss4 Structured version   Visualization version   GIF version

Theorem dfcoss4 34716
Description: Alternate definition of the class of cosets by 𝑅 (cf. the comment of df-coss 34712). (Contributed by Peter Mazsa, 12-Jul-2021.)
Assertion
Ref Expression
dfcoss4 𝑅 = ran (𝑅𝑅)

Proof of Theorem dfcoss4
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-coss 34712 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
2 rnxrn 34699 . 2 ran (𝑅𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
31, 2eqtr4i 2852 1 𝑅 = ran (𝑅𝑅)
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1656  wex 1878   class class class wbr 4875  {copab 4937  ran crn 5347  cxrn 34518  ccoss 34519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fo 6133  df-fv 6135  df-1st 7433  df-2nd 7434  df-ec 8016  df-xrn 34676  df-coss 34712
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator