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 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1o , 𝑦 ⟩ } )
Assertion	xpsfrn	⊢ ran 𝐹 = X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 )

Proof

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