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Theorem rncossdmcoss 36963
Description: The range of cosets is the domain of them (this should be rncoss 5928 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 36942 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
21el2v 3452 . . . 4 (𝑦𝑅𝑥𝑥𝑅𝑦)
32exbii 1851 . . 3 (∃𝑦 𝑦𝑅𝑥 ↔ ∃𝑦 𝑥𝑅𝑦)
43abbii 2803 . 2 {𝑥 ∣ ∃𝑦 𝑦𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
5 dfrn2 5845 . 2 ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦𝑅𝑥}
6 df-dm 5644 . 2 dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
74, 5, 63eqtr4i 2771 1 ran ≀ 𝑅 = dom ≀ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wex 1782  {cab 2710  Vcvv 3444   class class class wbr 5106  dom cdm 5634  ran crn 5635  ccoss 36680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645  df-coss 36919
This theorem is referenced by:  refrelcoss3  36971
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