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Metamath Proof Explorer

Theorem caovcld

Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014)

Ref Expression
Hypotheses caovclg.1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ) → ( 𝑥 𝐹 𝑦 ) ∈ 𝐸 )
caovcld.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
caovcld.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
Assertion caovcld ( 𝜑 → ( 𝐴 𝐹 𝐵 ) ∈ 𝐸 )

Proof

Step Hyp Ref Expression
1 caovclg.1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ) → ( 𝑥 𝐹 𝑦 ) ∈ 𝐸 )
2 caovcld.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
3 caovcld.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
4 id ⊢ ( 𝜑 → 𝜑 )
5 1 caovclg ⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐸 )
6 4 2 3 5 syl12anc ⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) ∈ 𝐸 )