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Theorem rncossdmcoss 38456
Description: The range of cosets is the domain of them (this should be rncoss 5986 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 38435 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
21el2v 3487 . . . 4 (𝑦𝑅𝑥𝑥𝑅𝑦)
32exbii 1848 . . 3 (∃𝑦 𝑦𝑅𝑥 ↔ ∃𝑦 𝑥𝑅𝑦)
43abbii 2809 . 2 {𝑥 ∣ ∃𝑦 𝑦𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
5 dfrn2 5899 . 2 ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦𝑅𝑥}
6 df-dm 5695 . 2 dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
74, 5, 63eqtr4i 2775 1 ran ≀ 𝑅 = dom ≀ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wex 1779  {cab 2714  Vcvv 3480   class class class wbr 5143  dom cdm 5685  ran crn 5686  ccoss 38182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-coss 38412
This theorem is referenced by:  refrelcoss3  38464
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