Theorem dmmpogaOLD

Description: Obsolete version of dmmpoga as of 29-May-2024. (Contributed by Alexander van der Vekens, 10-Feb-2019) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	dmmpog.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
Assertion	dmmpogaOLD	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = ( 𝐴 × 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		dmmpog.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
2	1	fnmpo	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn ( 𝐴 × 𝐵 ) )
3		fndm	⊢ ( 𝐹 Fn ( 𝐴 × 𝐵 ) → dom 𝐹 = ( 𝐴 × 𝐵 ) )
4	2 3	syl	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = ( 𝐴 × 𝐵 ) )