Theorem dmcoss3 35685

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 35654	. . 3 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)
2	1	dmeqi 5766	. 2 ⊢ dom ≀ 𝑅 = dom (𝑅 ∘ ◡𝑅)
3		rncnv 35550	. . . 4 ⊢ ran ◡𝑅 = dom 𝑅
4	3	eqimssi 4023	. . 3 ⊢ ran ◡𝑅 ⊆ dom 𝑅
5		dmcosseq 5837	. . 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 2842	1 ⊢ dom ≀ 𝑅 = dom ◡𝑅

Syntax hints:   = wceq 1531   ⊆ wss 3934  ^◡ccnv 5547  dom cdm 5548  ran crn 5549   ∘ ccom 5552   ≀ ccoss 35445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-coss 35651

This theorem is referenced by:  dmcoss2  35686  eldmcoss  35690