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Metamath Proof Explorer

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 ) )