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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 ⊢ ( 𝜑 → 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 )
Assertion fovcld ( ( 𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 )

Proof

Step Hyp Ref Expression
1 fovcld.1 ⊢ ( 𝜑 → 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 )
2 3simpc ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) )
3 ffnov ⊢ ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 ↔ ( 𝐹 Fn ( 𝑅 × 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ) )
4 3 simprbi ⊢ ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 → ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 )
5 1 4 syl ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 )
6 5 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ( 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 ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 ) )
12 2 6 11 sylc ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 )