Metamath Proof Explorer


Theorem fovcld

Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007) (Revised by Thierry Arnoux, 17-Feb-2017)

Ref Expression
Hypothesis fovcld.1 ( 𝜑𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 )
Assertion fovcld ( ( 𝜑𝐴𝑅𝐵𝑆 ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 )

Proof

Step Hyp Ref Expression
1 fovcld.1 ( 𝜑𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 )
2 3simpc ( ( 𝜑𝐴𝑅𝐵𝑆 ) → ( 𝐴𝑅𝐵𝑆 ) )
3 ffnov ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 ↔ ( 𝐹 Fn ( 𝑅 × 𝑆 ) ∧ ∀ 𝑥𝑅𝑦𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ) )
4 3 simprbi ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 → ∀ 𝑥𝑅𝑦𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 )
5 1 4 syl ( 𝜑 → ∀ 𝑥𝑅𝑦𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 )
6 5 3ad2ant1 ( ( 𝜑𝐴𝑅𝐵𝑆 ) → ∀ 𝑥𝑅𝑦𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 )
7 oveq1 ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑦 ) )
8 7 eleq1d ( 𝑥 = 𝐴 → ( ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ↔ ( 𝐴 𝐹 𝑦 ) ∈ 𝐶 ) )
9 oveq2 ( 𝑦 = 𝐵 → ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐵 ) )
10 9 eleq1d ( 𝑦 = 𝐵 → ( ( 𝐴 𝐹 𝑦 ) ∈ 𝐶 ↔ ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 ) )
11 8 10 rspc2v ( ( 𝐴𝑅𝐵𝑆 ) → ( ∀ 𝑥𝑅𝑦𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 ) )
12 2 6 11 sylc ( ( 𝜑𝐴𝑅𝐵𝑆 ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 )