Metamath Proof Explorer

Theorem tposresg

Description: The transposition restricted to a set. (Contributed by Zhi Wang, 6-Oct-2025)

Ref Expression
Assertion tposresg 
|- ( tpos F |` R ) = ( ( tpos F |` `' `' R ) u. ( F |` ( R i^i { (/) } ) ) )

Proof

Step Hyp Ref Expression
1 rescom 
 |-  ( ( tpos F |` ( ( _V X. _V ) u. { (/) } ) ) |` R ) = ( ( tpos F |` R ) |` ( ( _V X. _V ) u. { (/) } ) )
2 reltpos 
 |-  Rel tpos F
3 dmtposss 
 |-  dom tpos F C_ ( ( _V X. _V ) u. { (/) } )
4 relssres 
 |-  ( ( Rel tpos F /\ dom tpos F C_ ( ( _V X. _V ) u. { (/) } ) ) -> ( tpos F |` ( ( _V X. _V ) u. { (/) } ) ) = tpos F )
5 2 3 4 mp2an 
 |-  ( tpos F |` ( ( _V X. _V ) u. { (/) } ) ) = tpos F
6 5 reseq1i 
 |-  ( ( tpos F |` ( ( _V X. _V ) u. { (/) } ) ) |` R ) = ( tpos F |` R )
7 resres 
 |-  ( ( tpos F |` R ) |` ( ( _V X. _V ) u. { (/) } ) ) = ( tpos F |` ( R i^i ( ( _V X. _V ) u. { (/) } ) ) )
8 1 6 7 3eqtr3i 
 |-  ( tpos F |` R ) = ( tpos F |` ( R i^i ( ( _V X. _V ) u. { (/) } ) ) )
9 indi 
 |-  ( R i^i ( ( _V X. _V ) u. { (/) } ) ) = ( ( R i^i ( _V X. _V ) ) u. ( R i^i { (/) } ) )
10 cnvcnv 
 |-  `' `' R = ( R i^i ( _V X. _V ) )
11 10 uneq1i 
 |-  ( `' `' R u. ( R i^i { (/) } ) ) = ( ( R i^i ( _V X. _V ) ) u. ( R i^i { (/) } ) )
12 9 11 eqtr4i 
 |-  ( R i^i ( ( _V X. _V ) u. { (/) } ) ) = ( `' `' R u. ( R i^i { (/) } ) )
13 12 reseq2i 
 |-  ( tpos F |` ( R i^i ( ( _V X. _V ) u. { (/) } ) ) ) = ( tpos F |` ( `' `' R u. ( R i^i { (/) } ) ) )
14 resundi 
 |-  ( tpos F |` ( `' `' R u. ( R i^i { (/) } ) ) ) = ( ( tpos F |` `' `' R ) u. ( tpos F |` ( R i^i { (/) } ) ) )
15 rescom 
 |-  ( ( tpos F |` { (/) } ) |` R ) = ( ( tpos F |` R ) |` { (/) } )
16 tposres0 
 |-  ( tpos F |` { (/) } ) = ( F |` { (/) } )
17 16 reseq1i 
 |-  ( ( tpos F |` { (/) } ) |` R ) = ( ( F |` { (/) } ) |` R )
18 resres 
 |-  ( ( tpos F |` R ) |` { (/) } ) = ( tpos F |` ( R i^i { (/) } ) )
19 15 17 18 3eqtr3ri 
 |-  ( tpos F |` ( R i^i { (/) } ) ) = ( ( F |` { (/) } ) |` R )
20 rescom 
 |-  ( ( F |` { (/) } ) |` R ) = ( ( F |` R ) |` { (/) } )
21 resres 
 |-  ( ( F |` R ) |` { (/) } ) = ( F |` ( R i^i { (/) } ) )
22 19 20 21 3eqtri 
 |-  ( tpos F |` ( R i^i { (/) } ) ) = ( F |` ( R i^i { (/) } ) )
23 22 uneq2i 
 |-  ( ( tpos F |` `' `' R ) u. ( tpos F |` ( R i^i { (/) } ) ) ) = ( ( tpos F |` `' `' R ) u. ( F |` ( R i^i { (/) } ) ) )
24 14 23 eqtri 
 |-  ( tpos F |` ( `' `' R u. ( R i^i { (/) } ) ) ) = ( ( tpos F |` `' `' R ) u. ( F |` ( R i^i { (/) } ) ) )
25 8 13 24 3eqtri 
 |-  ( tpos F |` R ) = ( ( tpos F |` `' `' R ) u. ( F |` ( R i^i { (/) } ) ) )