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Metamath Proof Explorer

Theorem coemptyd

Description: Deduction about composition of classes with no relational content in common. (Contributed by RP, 24-Dec-2019)

Ref Expression
Hypothesis coemptyd.1 ⊢ ( 𝜑 → ( dom 𝐴 ∩ ran 𝐵 ) = ∅ )
Assertion coemptyd ( 𝜑 → ( 𝐴 ∘ 𝐵 ) = ∅ )

Proof

Step Hyp Ref Expression
1 coemptyd.1 ⊢ ( 𝜑 → ( dom 𝐴 ∩ ran 𝐵 ) = ∅ )
2 coeq0 ⊢ ( ( 𝐴 ∘ 𝐵 ) = ∅ ↔ ( dom 𝐴 ∩ ran 𝐵 ) = ∅ )
3 1 2 sylibr ⊢ ( 𝜑 → ( 𝐴 ∘ 𝐵 ) = ∅ )