Metamath Proof Explorer


Theorem f1osng

Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013)

Ref Expression
Assertion f1osng ( ( 𝐴𝑉𝐵𝑊 ) → { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1-onto→ { 𝐵 } )

Proof

Step Hyp Ref Expression
1 sneq ( 𝑎 = 𝐴 → { 𝑎 } = { 𝐴 } )
2 1 f1oeq2d ( 𝑎 = 𝐴 → ( { ⟨ 𝑎 , 𝑏 ⟩ } : { 𝑎 } –1-1-onto→ { 𝑏 } ↔ { ⟨ 𝑎 , 𝑏 ⟩ } : { 𝐴 } –1-1-onto→ { 𝑏 } ) )
3 opeq1 ( 𝑎 = 𝐴 → ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝑏 ⟩ )
4 3 sneqd ( 𝑎 = 𝐴 → { ⟨ 𝑎 , 𝑏 ⟩ } = { ⟨ 𝐴 , 𝑏 ⟩ } )
5 4 f1oeq1d ( 𝑎 = 𝐴 → ( { ⟨ 𝑎 , 𝑏 ⟩ } : { 𝐴 } –1-1-onto→ { 𝑏 } ↔ { ⟨ 𝐴 , 𝑏 ⟩ } : { 𝐴 } –1-1-onto→ { 𝑏 } ) )
6 2 5 bitrd ( 𝑎 = 𝐴 → ( { ⟨ 𝑎 , 𝑏 ⟩ } : { 𝑎 } –1-1-onto→ { 𝑏 } ↔ { ⟨ 𝐴 , 𝑏 ⟩ } : { 𝐴 } –1-1-onto→ { 𝑏 } ) )
7 sneq ( 𝑏 = 𝐵 → { 𝑏 } = { 𝐵 } )
8 7 f1oeq3d ( 𝑏 = 𝐵 → ( { ⟨ 𝐴 , 𝑏 ⟩ } : { 𝐴 } –1-1-onto→ { 𝑏 } ↔ { ⟨ 𝐴 , 𝑏 ⟩ } : { 𝐴 } –1-1-onto→ { 𝐵 } ) )
9 opeq2 ( 𝑏 = 𝐵 → ⟨ 𝐴 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
10 9 sneqd ( 𝑏 = 𝐵 → { ⟨ 𝐴 , 𝑏 ⟩ } = { ⟨ 𝐴 , 𝐵 ⟩ } )
11 10 f1oeq1d ( 𝑏 = 𝐵 → ( { ⟨ 𝐴 , 𝑏 ⟩ } : { 𝐴 } –1-1-onto→ { 𝐵 } ↔ { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1-onto→ { 𝐵 } ) )
12 8 11 bitrd ( 𝑏 = 𝐵 → ( { ⟨ 𝐴 , 𝑏 ⟩ } : { 𝐴 } –1-1-onto→ { 𝑏 } ↔ { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1-onto→ { 𝐵 } ) )
13 vex 𝑎 ∈ V
14 vex 𝑏 ∈ V
15 13 14 f1osn { ⟨ 𝑎 , 𝑏 ⟩ } : { 𝑎 } –1-1-onto→ { 𝑏 }
16 6 12 15 vtocl2g ( ( 𝐴𝑉𝐵𝑊 ) → { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1-onto→ { 𝐵 } )