Theorem dmcoss3 38055

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 38016	. . 3 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)
2	1	dmeqi 5907	. 2 ⊢ dom ≀ 𝑅 = dom (𝑅 ∘ ◡𝑅)
3		rncnv 37902	. . . 4 ⊢ ran ◡𝑅 = dom 𝑅
4	3	eqimssi 4037	. . 3 ⊢ ran ◡𝑅 ⊆ dom 𝑅
5		dmcosseq 5976	. . 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 2753	1 ⊢ dom ≀ 𝑅 = dom ◡𝑅

Syntax hints:   = wceq 1533   ⊆ wss 3944  ^◡ccnv 5677  dom cdm 5678  ran crn 5679   ∘ ccom 5682   ≀ ccoss 37779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-coss 38013

This theorem is referenced by:  dmcoss2  38056  eldmcoss  38060