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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 
|- ( ( A e. V /\ B e. W ) -> { <. 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 e. _V
14 vex 
 |-  b e. _V
15 13 14 f1osn 
 |-  { <. a , b >. } : { a } -1-1-onto-> { b }
16 6 12 15 vtocl2g 
 |-  ( ( A e. V /\ B e. W ) -> { <. A , B >. } : { A } -1-1-onto-> { B } )