This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cnvcnvsn

Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn , this does not need any sethood assumptions on A and B .) (Contributed by Mario Carneiro, 26-Apr-2015)

Ref Expression
Assertion cnvcnvsn 
|- `' `' { <. A , B >. } = `' { <. B , A >. }

Proof

Step Hyp Ref Expression
1 relcnv 
 |-  Rel `' `' { <. A , B >. }
2 relcnv 
 |-  Rel `' { <. B , A >. }
3 vex 
 |-  x e. _V
4 vex 
 |-  y e. _V
5 3 4 opelcnv 
 |-  ( <. x , y >. e. `' `' { <. A , B >. } <-> <. y , x >. e. `' { <. A , B >. } )
6 ancom 
 |-  ( ( x = A /\ y = B ) <-> ( y = B /\ x = A ) )
7 3 4 opth 
 |-  ( <. x , y >. = <. A , B >. <-> ( x = A /\ y = B ) )
8 4 3 opth 
 |-  ( <. y , x >. = <. B , A >. <-> ( y = B /\ x = A ) )
9 6 7 8 3bitr4i 
 |-  ( <. x , y >. = <. A , B >. <-> <. y , x >. = <. B , A >. )
10 opex 
 |-  <. x , y >. e. _V
11 10 elsn 
 |-  ( <. x , y >. e. { <. A , B >. } <-> <. x , y >. = <. A , B >. )
12 opex 
 |-  <. y , x >. e. _V
13 12 elsn 
 |-  ( <. y , x >. e. { <. B , A >. } <-> <. y , x >. = <. B , A >. )
14 9 11 13 3bitr4i 
 |-  ( <. x , y >. e. { <. A , B >. } <-> <. y , x >. e. { <. B , A >. } )
15 4 3 opelcnv 
 |-  ( <. y , x >. e. `' { <. A , B >. } <-> <. x , y >. e. { <. A , B >. } )
16 3 4 opelcnv 
 |-  ( <. x , y >. e. `' { <. B , A >. } <-> <. y , x >. e. { <. B , A >. } )
17 14 15 16 3bitr4i 
 |-  ( <. y , x >. e. `' { <. A , B >. } <-> <. x , y >. e. `' { <. B , A >. } )
18 5 17 bitri 
 |-  ( <. x , y >. e. `' `' { <. A , B >. } <-> <. x , y >. e. `' { <. B , A >. } )
19 1 2 18 eqrelriiv 
 |-  `' `' { <. A , B >. } = `' { <. B , A >. }