Theorem dmcoss3 39667

Description: The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.)

Assertion

Ref	Expression
dmcoss3	⊢ dom ≀ 𝑅 = dom ◡𝑅

Proof of Theorem dmcoss3

Step	Hyp	Ref	Expression
1		dfcoss3 39628	. . 3 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)
2	1	dmeqi 5893	. 2 ⊢ dom ≀ 𝑅 = dom (𝑅 ∘ ◡𝑅)
3		rncnv 39430	. . . 4 ⊢ ran ◡𝑅 = dom 𝑅
4	3	eqimssi 3973	. . 3 ⊢ ran ◡𝑅 ⊆ dom 𝑅
5		dmcosseq 5967	. . 3 ⊢ (ran ◡𝑅 ⊆ dom 𝑅 → dom (𝑅 ∘ ◡𝑅) = dom ◡𝑅)
6	4, 5	ax-mp 5	. 2 ⊢ dom (𝑅 ∘ ◡𝑅) = dom ◡𝑅
7	2, 6	eqtri 2786	1 ⊢ dom ≀ 𝑅 = dom ◡𝑅

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3900  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ∘ ccom 5659   ≀ ccoss 39307

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-coss 39625

This theorem is used by:  dmcoss2  39668  eldmcoss  39672