Metamath Proof Explorer


Theorem mptcnv

Description: The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017)

Ref Expression
Hypothesis mptcnv.1 φxAy=ByCx=D
Assertion mptcnv φxAB-1=yCD

Proof

Step Hyp Ref Expression
1 mptcnv.1 φxAy=ByCx=D
2 1 opabbidv φyx|xAy=B=yx|yCx=D
3 df-mpt xAB=xy|xAy=B
4 3 cnveqi xAB-1=xy|xAy=B-1
5 cnvopab xy|xAy=B-1=yx|xAy=B
6 4 5 eqtri xAB-1=yx|xAy=B
7 df-mpt yCD=yx|yCx=D
8 2 6 7 3eqtr4g φxAB-1=yCD