Theorem rncossdmcoss 38880

Description: The range of cosets is the domain of them (this should be rncoss 5926 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 38859	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦))
2	1	el2v 3437	. . . 4 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)
3	2	exbii 1850	. . 3 ⊢ (∃𝑦 𝑦 ≀ 𝑅𝑥 ↔ ∃𝑦 𝑥 ≀ 𝑅𝑦)
4	3	abbii 2804	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦}
5		dfrn2 5837	. 2 ⊢ ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥}
6		df-dm 5634	. 2 ⊢ dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦}
7	4, 5, 6	3eqtr4i 2770	1 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅

Syntax hints:   ↔ wb 206   = wceq 1542  ∃wex 1781  {cab 2715  Vcvv 3430   class class class wbr 5086  dom cdm 5624  ran crn 5625   ≀ ccoss 38518

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5632  df-dm 5634  df-rn 5635  df-coss 38836

This theorem is referenced by:  refrelcoss3  38888