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Metamath Proof Explorer

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 )