Metamath Proof Explorer


Theorem tposf

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

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

Proof

Step Hyp Ref Expression
1 relxp
 |-  Rel ( A X. B )
2 tposf2
 |-  ( Rel ( A X. B ) -> ( F : ( A X. B ) --> C -> tpos F : `' ( A X. B ) --> C ) )
3 1 2 ax-mp
 |-  ( F : ( A X. B ) --> C -> tpos F : `' ( A X. B ) --> C )
4 cnvxp
 |-  `' ( A X. B ) = ( B X. A )
5 4 feq2i
 |-  ( tpos F : `' ( A X. B ) --> C <-> tpos F : ( B X. A ) --> C )
6 3 5 sylib
 |-  ( F : ( A X. B ) --> C -> tpos F : ( B X. A ) --> C )