Theorem dmcoss3 39443

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 39404	. . 3 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)
2	1	dmeqi 5886	. 2 ⊢ dom ≀ 𝑅 = dom (𝑅 ∘ ◡𝑅)
3		rncnv 39206	. . . 4 ⊢ ran ◡𝑅 = dom 𝑅
4	3	eqimssi 3991	. . 3 ⊢ ran ◡𝑅 ⊆ dom 𝑅
5		dmcosseq 5960	. . 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 2784	1 ⊢ dom ≀ 𝑅 = dom ◡𝑅

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3899  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ∘ ccom 5655   ≀ ccoss 39083

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-coss 39401

This theorem is used by:  dmcoss2  39444  eldmcoss  39448