Theorem rncossdmcoss 37864

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 37843	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦))
2	1	el2v 3477	. . . 4 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)
3	2	exbii 1843	. . 3 ⊢ (∃𝑦 𝑦 ≀ 𝑅𝑥 ↔ ∃𝑦 𝑥 ≀ 𝑅𝑦)
4	3	abbii 2797	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦}
5		dfrn2 5885	. 2 ⊢ ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥}
6		df-dm 5682	. 2 ⊢ dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦}
7	4, 5, 6	3eqtr4i 2765	1 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅

Syntax hints:   ↔ wb 205   = wceq 1534  ∃wex 1774  {cab 2704  Vcvv 3469   class class class wbr 5142  dom cdm 5672  ran crn 5673   ≀ ccoss 37583

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 37820

This theorem is referenced by:  refrelcoss3  37872