Theorem cotr3

Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020)

	Ref	Expression
Hypotheses	cotr3.a	⊢ 𝐴 = dom 𝑅
	cotr3.b	⊢ 𝐵 = ( 𝐴 ∩ 𝐶 )
	cotr3.c	⊢ 𝐶 = ran 𝑅
Assertion	cotr3	⊢ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		cotr3.a	⊢ 𝐴 = dom 𝑅
2		cotr3.b	⊢ 𝐵 = ( 𝐴 ∩ 𝐶 )
3		cotr3.c	⊢ 𝐶 = ran 𝑅
4	1	eqimss2i	⊢ dom 𝑅 ⊆ 𝐴
5	1 3	ineq12i	⊢ ( 𝐴 ∩ 𝐶 ) = ( dom 𝑅 ∩ ran 𝑅 )
6	2 5	eqtri	⊢ 𝐵 = ( dom 𝑅 ∩ ran 𝑅 )
7	6	eqimss2i	⊢ ( dom 𝑅 ∩ ran 𝑅 ) ⊆ 𝐵
8	3	eqimss2i	⊢ ran 𝑅 ⊆ 𝐶
9	4 7 8	cotr2	⊢ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) )