Theorem mptcnv

Description: The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017)

	Ref	Expression
Hypothesis	mptcnv.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷 ) ) )
Assertion	mptcnv	⊢ ( 𝜑 → ◡ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐶 ↦ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		mptcnv.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷 ) ) )
2	1	opabbidv	⊢ ( 𝜑 → { ⟨ 𝑦 , 𝑥 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) } = { ⟨ 𝑦 , 𝑥 ⟩ ∣ ( 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷 ) } )
3		df-mpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) }
4	3	cnveqi	⊢ ◡ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) }
5		cnvopab	⊢ ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) } = { ⟨ 𝑦 , 𝑥 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) }
6	4 5	eqtri	⊢ ◡ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = { ⟨ 𝑦 , 𝑥 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) }
7		df-mpt	⊢ ( 𝑦 ∈ 𝐶 ↦ 𝐷 ) = { ⟨ 𝑦 , 𝑥 ⟩ ∣ ( 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷 ) }
8	2 6 7	3eqtr4g	⊢ ( 𝜑 → ◡ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐶 ↦ 𝐷 ) )