dmcoss3 - Mathbox for Peter Mazsa

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Theorem dmcoss3 37925

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 37886	. . 3 ⊢ ≀ 𝑅 = (𝑅 ∘ ^◡𝑅)
2	1	dmeqi 5907	. 2 ⊢ dom ≀ 𝑅 = dom (𝑅 ∘ ^◡𝑅)
3		rncnv 37772	. . . 4 ⊢ ran ^◡𝑅 = dom 𝑅
4	3	eqimssi 4040	. . 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 2756	1 ⊢ dom ≀ 𝑅 = dom ^◡𝑅

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ⊆ wss 3947 ^◡ccnv 5677 dom cdm 5678 ran crn 5679 ∘ ccom 5682 ≀ ccoss 37648

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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-coss 37883

This theorem is referenced by: dmcoss2 37926 eldmcoss 37930

	
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