Theorem dmcoeq

Description: Domain of a composition. (Contributed by NM, 19-Mar-1998)

	Ref	Expression
Assertion	dmcoeq	⊢ ( dom 𝐴 = ran 𝐵 → dom ( 𝐴 ∘ 𝐵 ) = dom 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eqimss2	⊢ ( dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴 )
2		dmcosseq	⊢ ( ran 𝐵 ⊆ dom 𝐴 → dom ( 𝐴 ∘ 𝐵 ) = dom 𝐵 )
3	1 2	syl	⊢ ( dom 𝐴 = ran 𝐵 → dom ( 𝐴 ∘ 𝐵 ) = dom 𝐵 )