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Theorem rncossdmcoss 37979
Description: The range of cosets is the domain of them (this should be rncoss 5970 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.)
Assertion
Ref Expression
rncossdmcoss ran ≀ 𝑅 = dom ≀ 𝑅

Proof of Theorem rncossdmcoss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brcosscnvcoss 37958 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
21el2v 3471 . . . 4 (𝑦𝑅𝑥𝑥𝑅𝑦)
32exbii 1842 . . 3 (∃𝑦 𝑦𝑅𝑥 ↔ ∃𝑦 𝑥𝑅𝑦)
43abbii 2795 . 2 {𝑥 ∣ ∃𝑦 𝑦𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
5 dfrn2 5886 . 2 ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦𝑅𝑥}
6 df-dm 5683 . 2 dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
74, 5, 63eqtr4i 2763 1 ran ≀ 𝑅 = dom ≀ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wex 1773  {cab 2702  Vcvv 3463   class class class wbr 5144  dom cdm 5673  ran crn 5674  ccoss 37701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-cnv 5681  df-dm 5683  df-rn 5684  df-coss 37935
This theorem is referenced by:  refrelcoss3  37987
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