Theorem brelrng

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008)

	Ref	Expression
Assertion	brelrng	⊢ ( ( 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴 𝐶 𝐵 ) → 𝐵 ∈ ran 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		brcnvg	⊢ ( ( 𝐵 ∈ 𝐺 ∧ 𝐴 ∈ 𝐹 ) → ( 𝐵 ◡ 𝐶 𝐴 ↔ 𝐴 𝐶 𝐵 ) )
2	1	ancoms	⊢ ( ( 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ) → ( 𝐵 ◡ 𝐶 𝐴 ↔ 𝐴 𝐶 𝐵 ) )
3	2	biimp3ar	⊢ ( ( 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴 𝐶 𝐵 ) → 𝐵 ◡ 𝐶 𝐴 )
4		breldmg	⊢ ( ( 𝐵 ∈ 𝐺 ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ◡ 𝐶 𝐴 ) → 𝐵 ∈ dom ◡ 𝐶 )
5	4	3com12	⊢ ( ( 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐵 ◡ 𝐶 𝐴 ) → 𝐵 ∈ dom ◡ 𝐶 )
6	3 5	syld3an3	⊢ ( ( 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴 𝐶 𝐵 ) → 𝐵 ∈ dom ◡ 𝐶 )
7		df-rn	⊢ ran 𝐶 = dom ◡ 𝐶
8	6 7	eleqtrrdi	⊢ ( ( 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴 𝐶 𝐵 ) → 𝐵 ∈ ran 𝐶 )