Metamath Proof Explorer


Theorem supppreima

Description: Express the support of a function as the preimage of its range except zero. (Contributed by Thierry Arnoux, 24-Jun-2024)

Ref Expression
Assertion supppreima Fun F F V Z W F supp Z = F -1 ran F Z

Proof

Step Hyp Ref Expression
1 cnvimarndm F -1 ran F = dom F
2 1 a1i Fun F F V Z W F -1 ran F = dom F
3 2 difeq1d Fun F F V Z W F -1 ran F F -1 Z = dom F F -1 Z
4 difpreima Fun F F -1 ran F Z = F -1 ran F F -1 Z
5 4 3ad2ant1 Fun F F V Z W F -1 ran F Z = F -1 ran F F -1 Z
6 suppssdm F supp Z dom F
7 dfss4 F supp Z dom F dom F dom F supp Z F = F supp Z
8 6 7 mpbi dom F dom F supp Z F = F supp Z
9 suppiniseg Fun F F V Z W dom F supp Z F = F -1 Z
10 9 difeq2d Fun F F V Z W dom F dom F supp Z F = dom F F -1 Z
11 8 10 eqtr3id Fun F F V Z W F supp Z = dom F F -1 Z
12 3 5 11 3eqtr4rd Fun F F V Z W F supp Z = F -1 ran F Z