Metamath Proof Explorer


Theorem xpsfrn

Description: A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015)

Ref Expression
Hypothesis xpsff1o.f 𝐹 = ( 𝑥𝐴 , 𝑦𝐵 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1o , 𝑦 ⟩ } )
Assertion xpsfrn ran 𝐹 = X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 )

Proof

Step Hyp Ref Expression
1 xpsff1o.f 𝐹 = ( 𝑥𝐴 , 𝑦𝐵 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1o , 𝑦 ⟩ } )
2 1 xpsff1o 𝐹 : ( 𝐴 × 𝐵 ) –1-1-ontoX 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 )
3 f1ofo ( 𝐹 : ( 𝐴 × 𝐵 ) –1-1-ontoX 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) → 𝐹 : ( 𝐴 × 𝐵 ) –ontoX 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) )
4 forn ( 𝐹 : ( 𝐴 × 𝐵 ) –ontoX 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) → ran 𝐹 = X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) )
5 2 3 4 mp2b ran 𝐹 = X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 )