Theorem reldmmpo

Description: The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014)

	Ref	Expression
Hypothesis	rngop.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
Assertion	reldmmpo	⊢ Rel dom 𝐹

Proof

Step	Hyp	Ref	Expression
1		rngop.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
2		reldmoprab	⊢ Rel dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐶 ) }
3		df-mpo	⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐶 ) }
4	1 3	eqtri	⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐶 ) }
5	4	dmeqi	⊢ dom 𝐹 = dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐶 ) }
6	5	releqi	⊢ ( Rel dom 𝐹 ↔ Rel dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐶 ) } )
7	2 6	mpbir	⊢ Rel dom 𝐹