Theorem rncossdmcoss 38491

Description: The range of cosets is the domain of them (this should be rncoss 5916 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.

Step	Hyp	Ref	Expression
1		brcosscnvcoss 38470	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦))
2	1	el2v 3443	. . . 4 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)
3	2	exbii 1849	. . 3 ⊢ (∃𝑦 𝑦 ≀ 𝑅𝑥 ↔ ∃𝑦 𝑥 ≀ 𝑅𝑦)
4	3	abbii 2798	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦}
5		dfrn2 5828	. 2 ⊢ ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥}
6		df-dm 5626	. 2 ⊢ dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦}
7	4, 5, 6	3eqtr4i 2764	1 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅

Syntax hints:   ↔ wb 206   = wceq 1541  ∃wex 1780  {cab 2709  Vcvv 3436   class class class wbr 5091  dom cdm 5616  ran crn 5617   ≀ ccoss 38214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-cnv 5624  df-dm 5626  df-rn 5627  df-coss 38447

This theorem is referenced by:  refrelcoss3  38499