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Theorem rncossdmcoss 37851
Description: The range of cosets is the domain of them (this should be rncoss 5969 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 37830 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
21el2v 3477 . . . 4 (𝑦𝑅𝑥𝑥𝑅𝑦)
32exbii 1843 . . 3 (∃𝑦 𝑦𝑅𝑥 ↔ ∃𝑦 𝑥𝑅𝑦)
43abbii 2797 . 2 {𝑥 ∣ ∃𝑦 𝑦𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
5 dfrn2 5885 . 2 ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦𝑅𝑥}
6 df-dm 5682 . 2 dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
74, 5, 63eqtr4i 2765 1 ran ≀ 𝑅 = dom ≀ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wex 1774  {cab 2704  Vcvv 3469   class class class wbr 5142  dom cdm 5672  ran crn 5673  ccoss 37570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-cnv 5680  df-dm 5682  df-rn 5683  df-coss 37807
This theorem is referenced by:  refrelcoss3  37859
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