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Metamath Proof Explorer

Theorem mapsnf1o

Description: A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015)

Ref Expression
Hypothesis ixpsnf1o.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( { 𝐼 } × { 𝑥 } ) )
Assertion mapsnf1o ( ( 𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝐹 : 𝐴 –1-1-onto→ ( 𝐴 ↑m { 𝐼 } ) )

Proof

Step Hyp Ref Expression
1 ixpsnf1o.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( { 𝐼 } × { 𝑥 } ) )
2 1 ixpsnf1o ⊢ ( 𝐼 ∈ 𝑊 → 𝐹 : 𝐴 –1-1-onto→ X 𝑦 ∈ { 𝐼 } 𝐴 )
3 2 adantl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝐹 : 𝐴 –1-1-onto→ X 𝑦 ∈ { 𝐼 } 𝐴 )
4 snex ⊢ { 𝐼 } ∈ V
5 ixpconstg ⊢ ( ( { 𝐼 } ∈ V ∧ 𝐴 ∈ 𝑉 ) → X 𝑦 ∈ { 𝐼 } 𝐴 = ( 𝐴 ↑m { 𝐼 } ) )
6 5 eqcomd ⊢ ( ( { 𝐼 } ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ↑m { 𝐼 } ) = X 𝑦 ∈ { 𝐼 } 𝐴 )
7 4 6 mpan ⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ↑m { 𝐼 } ) = X 𝑦 ∈ { 𝐼 } 𝐴 )
8 7 adantr ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → ( 𝐴 ↑m { 𝐼 } ) = X 𝑦 ∈ { 𝐼 } 𝐴 )
9 8 f1oeq3d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → ( 𝐹 : 𝐴 –1-1-onto→ ( 𝐴 ↑m { 𝐼 } ) ↔ 𝐹 : 𝐴 –1-1-onto→ X 𝑦 ∈ { 𝐼 } 𝐴 ) )
10 3 9 mpbird ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝐹 : 𝐴 –1-1-onto→ ( 𝐴 ↑m { 𝐼 } ) )