Theorem tposmpo

Description: Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015)

	Ref	Expression
Hypothesis	tposmpo.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
Assertion	tposmpo	⊢ tpos 𝐹 = ( 𝑦 ∈ 𝐵 , 𝑥 ∈ 𝐴 ↦ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		tposmpo.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
2		df-mpo	⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐶 ) }
3		ancom	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) )
4	3	anbi1i	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐶 ) ↔ ( ( 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑧 = 𝐶 ) )
5	4	oprabbii	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐶 ) } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑧 = 𝐶 ) }
6	1 2 5	3eqtri	⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑧 = 𝐶 ) }
7	6	tposoprab	⊢ tpos 𝐹 = { ⟨ ⟨ 𝑦 , 𝑥 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑧 = 𝐶 ) }
8		df-mpo	⊢ ( 𝑦 ∈ 𝐵 , 𝑥 ∈ 𝐴 ↦ 𝐶 ) = { ⟨ ⟨ 𝑦 , 𝑥 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑧 = 𝐶 ) }
9	7 8	eqtr4i	⊢ tpos 𝐹 = ( 𝑦 ∈ 𝐵 , 𝑥 ∈ 𝐴 ↦ 𝐶 )