Metamath Proof Explorer


Theorem funcnvres

Description: The converse of a restricted function. (Contributed by NM, 27-Mar-1998)

Ref Expression
Assertion funcnvres Fun F -1 F A -1 = F -1 F A

Proof

Step Hyp Ref Expression
1 df-ima F A = ran F A
2 df-rn ran F A = dom F A -1
3 1 2 eqtri F A = dom F A -1
4 3 reseq2i F -1 F A = F -1 dom F A -1
5 resss F A F
6 cnvss F A F F A -1 F -1
7 5 6 ax-mp F A -1 F -1
8 funssres Fun F -1 F A -1 F -1 F -1 dom F A -1 = F A -1
9 7 8 mpan2 Fun F -1 F -1 dom F A -1 = F A -1
10 4 9 syl5req Fun F -1 F A -1 = F -1 F A