Theorem rncossdmcoss 34842

Description: The range of cosets is the domain of them (this should be rncoss 5634 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 34826	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦))
2	1	el2v 3404	. . . 4 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)
3	2	exbii 1892	. . 3 ⊢ (∃𝑦 𝑦 ≀ 𝑅𝑥 ↔ ∃𝑦 𝑥 ≀ 𝑅𝑦)
4	3	abbii 2908	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦}
5		dfrn2 5558	. 2 ⊢ ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥}
6		df-dm 5367	. 2 ⊢ dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦}
7	4, 5, 6	3eqtr4i 2812	1 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅

Syntax hints:   ↔ wb 198   = wceq 1601  ∃wex 1823  {cab 2763  Vcvv 3398   class class class wbr 4888  dom cdm 5357  ran crn 5358   ≀ ccoss 34615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-cnv 5365  df-dm 5367  df-rn 5368  df-coss 34806

This theorem is referenced by:  refrelcoss3  34850