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Theorem rncossdmcoss 35874
 Description: The range of cosets is the domain of them (this should be rncoss 5809 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 35858 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
21el2v 3448 . . . 4 (𝑦𝑅𝑥𝑥𝑅𝑦)
32exbii 1849 . . 3 (∃𝑦 𝑦𝑅𝑥 ↔ ∃𝑦 𝑥𝑅𝑦)
43abbii 2863 . 2 {𝑥 ∣ ∃𝑦 𝑦𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
5 dfrn2 5724 . 2 ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦𝑅𝑥}
6 df-dm 5530 . 2 dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
74, 5, 63eqtr4i 2831 1 ran ≀ 𝑅 = dom ≀ 𝑅
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  ∃wex 1781  {cab 2776  Vcvv 3441   class class class wbr 5031  dom cdm 5520  ran crn 5521   ≀ ccoss 35632 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-cnv 5528  df-dm 5530  df-rn 5531  df-coss 35838 This theorem is referenced by:  refrelcoss3  35882
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