Theorem rncossdmcoss 39175

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.

Step	Hyp	Ref	Expression
1		brcosscnvcoss 39154	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦))
2	1	el2v 3462	. . . 4 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)
3	2	exbii 1878	. . 3 ⊢ (∃𝑦 𝑦 ≀ 𝑅𝑥 ↔ ∃𝑦 𝑥 ≀ 𝑅𝑦)
4	3	abbii 2830	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦}
5		dfrn2 5880	. 2 ⊢ ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥}
6		df-dm 5673	. 2 ⊢ dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦}
7	4, 5, 6	3eqtr4i 2796	1 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅

Syntax hints:   ↔ wb 209   = wceq 1570  ∃wex 1809  {cab 2741  Vcvv 3455   class class class wbr 5110  dom cdm 5663  ran crn 5664   ≀ ccoss 38813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-cnv 5671  df-dm 5673  df-rn 5674  df-coss 39131

This theorem is referenced by:  refrelcoss3  39183