Theorem dmcoss3 38388

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 38349	. . 3 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)
2	1	dmeqi 5895	. 2 ⊢ dom ≀ 𝑅 = dom (𝑅 ∘ ◡𝑅)
3		rncnv 38235	. . . 4 ⊢ ran ◡𝑅 = dom 𝑅
4	3	eqimssi 4024	. . 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 2757	1 ⊢ dom ≀ 𝑅 = dom ◡𝑅

Syntax hints:   = wceq 1539   ⊆ wss 3931  ^◡ccnv 5664  dom cdm 5665  ran crn 5666   ∘ ccom 5669   ≀ ccoss 38116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-coss 38346

This theorem is referenced by:  dmcoss2  38389  eldmcoss  38393