Metamath Proof Explorer

Theorem tposfo

Description: The domain and range of a transposition. (Contributed by NM, 10-Sep-2015)

Ref Expression
Assertion tposfo 
|- ( F : ( A X. B ) -onto-> C -> tpos F : ( B X. A ) -onto-> C )

Proof

Step Hyp Ref Expression
1 relxp 
 |-  Rel ( A X. B )
2 tposfo2 
 |-  ( Rel ( A X. B ) -> ( F : ( A X. B ) -onto-> C -> tpos F : `' ( A X. B ) -onto-> C ) )
3 1 2 ax-mp 
 |-  ( F : ( A X. B ) -onto-> C -> tpos F : `' ( A X. B ) -onto-> C )
4 cnvxp 
 |-  `' ( A X. B ) = ( B X. A )
5 foeq2 
 |-  ( `' ( A X. B ) = ( B X. A ) -> ( tpos F : `' ( A X. B ) -onto-> C <-> tpos F : ( B X. A ) -onto-> C ) )
6 4 5 ax-mp 
 |-  ( tpos F : `' ( A X. B ) -onto-> C <-> tpos F : ( B X. A ) -onto-> C )
7 3 6 sylib 
 |-  ( F : ( A X. B ) -onto-> C -> tpos F : ( B X. A ) -onto-> C )