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Metamath Proof Explorer

Theorem brtpos

Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015)

Ref Expression
Assertion brtpos 
|- ( C e. V -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) )

Proof

Step Hyp Ref Expression
1 brtpos2 
 |-  ( C e. V -> ( <. A , B >. tpos F C <-> ( <. A , B >. e. ( `' dom F u. { (/) } ) /\ U. `' { <. A , B >. } F C ) ) )
2 1 adantr 
 |-  ( ( C e. V /\ ( A e. _V /\ B e. _V ) ) -> ( <. A , B >. tpos F C <-> ( <. A , B >. e. ( `' dom F u. { (/) } ) /\ U. `' { <. A , B >. } F C ) ) )
3 opex 
 |-  <. B , A >. e. _V
4 breldmg 
 |-  ( ( <. B , A >. e. _V /\ C e. V /\ <. B , A >. F C ) -> <. B , A >. e. dom F )
5 4 3expia 
 |-  ( ( <. B , A >. e. _V /\ C e. V ) -> ( <. B , A >. F C -> <. B , A >. e. dom F ) )
6 3 5 mpan 
 |-  ( C e. V -> ( <. B , A >. F C -> <. B , A >. e. dom F ) )
7 6 adantr 
 |-  ( ( C e. V /\ ( A e. _V /\ B e. _V ) ) -> ( <. B , A >. F C -> <. B , A >. e. dom F ) )
8 opelcnvg 
 |-  ( ( A e. _V /\ B e. _V ) -> ( <. A , B >. e. `' dom F <-> <. B , A >. e. dom F ) )
9 8 adantl 
 |-  ( ( C e. V /\ ( A e. _V /\ B e. _V ) ) -> ( <. A , B >. e. `' dom F <-> <. B , A >. e. dom F ) )
10 7 9 sylibrd 
 |-  ( ( C e. V /\ ( A e. _V /\ B e. _V ) ) -> ( <. B , A >. F C -> <. A , B >. e. `' dom F ) )
11 elun1 
 |-  ( <. A , B >. e. `' dom F -> <. A , B >. e. ( `' dom F u. { (/) } ) )
12 10 11 syl6 
 |-  ( ( C e. V /\ ( A e. _V /\ B e. _V ) ) -> ( <. B , A >. F C -> <. A , B >. e. ( `' dom F u. { (/) } ) ) )
13 12 pm4.71rd 
 |-  ( ( C e. V /\ ( A e. _V /\ B e. _V ) ) -> ( <. B , A >. F C <-> ( <. A , B >. e. ( `' dom F u. { (/) } ) /\ <. B , A >. F C ) ) )
14 opswap 
 |-  U. `' { <. A , B >. } = <. B , A >.
15 14 breq1i 
 |-  ( U. `' { <. A , B >. } F C <-> <. B , A >. F C )
16 15 anbi2i 
 |-  ( ( <. A , B >. e. ( `' dom F u. { (/) } ) /\ U. `' { <. A , B >. } F C ) <-> ( <. A , B >. e. ( `' dom F u. { (/) } ) /\ <. B , A >. F C ) )
17 13 16 bitr4di 
 |-  ( ( C e. V /\ ( A e. _V /\ B e. _V ) ) -> ( <. B , A >. F C <-> ( <. A , B >. e. ( `' dom F u. { (/) } ) /\ U. `' { <. A , B >. } F C ) ) )
18 2 17 bitr4d 
 |-  ( ( C e. V /\ ( A e. _V /\ B e. _V ) ) -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) )
19 18 ex 
 |-  ( C e. V -> ( ( A e. _V /\ B e. _V ) -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) ) )
20 brtpos0 
 |-  ( C e. V -> ( (/) tpos F C <-> (/) F C ) )
21 opprc 
 |-  ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
22 21 breq1d 
 |-  ( -. ( A e. _V /\ B e. _V ) -> ( <. A , B >. tpos F C <-> (/) tpos F C ) )
23 ancom 
 |-  ( ( A e. _V /\ B e. _V ) <-> ( B e. _V /\ A e. _V ) )
24 opprc 
 |-  ( -. ( B e. _V /\ A e. _V ) -> <. B , A >. = (/) )
25 24 breq1d 
 |-  ( -. ( B e. _V /\ A e. _V ) -> ( <. B , A >. F C <-> (/) F C ) )
26 23 25 sylnbi 
 |-  ( -. ( A e. _V /\ B e. _V ) -> ( <. B , A >. F C <-> (/) F C ) )
27 22 26 bibi12d 
 |-  ( -. ( A e. _V /\ B e. _V ) -> ( ( <. A , B >. tpos F C <-> <. B , A >. F C ) <-> ( (/) tpos F C <-> (/) F C ) ) )
28 20 27 syl5ibrcom 
 |-  ( C e. V -> ( -. ( A e. _V /\ B e. _V ) -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) ) )
29 19 28 pm2.61d 
 |-  ( C e. V -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) )