Theorem curry2f

Description: Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008)

	Ref	Expression
Hypothesis	curry2.1	⊢ 𝐺 = ( 𝐹 ∘ ◡ ( 1st ↾ ( V × { 𝐶 } ) ) )
Assertion	curry2f	⊢ ( ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐷 ∧ 𝐶 ∈ 𝐵 ) → 𝐺 : 𝐴 ⟶ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		curry2.1	⊢ 𝐺 = ( 𝐹 ∘ ◡ ( 1st ↾ ( V × { 𝐶 } ) ) )
2		ffn	⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐷 → 𝐹 Fn ( 𝐴 × 𝐵 ) )
3	1	curry2	⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ) → 𝐺 = ( 𝑥 ∈ 𝐴 ↦ ( 𝑥 𝐹 𝐶 ) ) )
4	2 3	sylan	⊢ ( ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐷 ∧ 𝐶 ∈ 𝐵 ) → 𝐺 = ( 𝑥 ∈ 𝐴 ↦ ( 𝑥 𝐹 𝐶 ) ) )
5		fovcdm	⊢ ( ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐷 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵 ) → ( 𝑥 𝐹 𝐶 ) ∈ 𝐷 )
6	5	3com23	⊢ ( ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐷 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 𝐹 𝐶 ) ∈ 𝐷 )
7	6	3expa	⊢ ( ( ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐷 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 𝐹 𝐶 ) ∈ 𝐷 )
8	4 7	fmpt3d	⊢ ( ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐷 ∧ 𝐶 ∈ 𝐵 ) → 𝐺 : 𝐴 ⟶ 𝐷 )