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Metamath Proof Explorer

Theorem dffix2

Description: The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012)

Ref Expression
Assertion dffix2 
|- Fix A = ran ( A i^i _I )

Proof

Step Hyp Ref Expression
1 vex 
 |-  x e. _V
2 1 elfix 
 |-  ( x e. Fix A <-> x A x )
3 1 elrn 
 |-  ( x e. ran ( A i^i _I ) <-> E. y y ( A i^i _I ) x )
4 brin 
 |-  ( y ( A i^i _I ) x <-> ( y A x /\ y _I x ) )
5 ancom 
 |-  ( ( y A x /\ y _I x ) <-> ( y _I x /\ y A x ) )
6 1 ideq 
 |-  ( y _I x <-> y = x )
7 6 anbi1i 
 |-  ( ( y _I x /\ y A x ) <-> ( y = x /\ y A x ) )
8 4 5 7 3bitri 
 |-  ( y ( A i^i _I ) x <-> ( y = x /\ y A x ) )
9 8 exbii 
 |-  ( E. y y ( A i^i _I ) x <-> E. y ( y = x /\ y A x ) )
10 breq1 
 |-  ( y = x -> ( y A x <-> x A x ) )
11 10 equsexvw 
 |-  ( E. y ( y = x /\ y A x ) <-> x A x )
12 3 9 11 3bitri 
 |-  ( x e. ran ( A i^i _I ) <-> x A x )
13 2 12 bitr4i 
 |-  ( x e. Fix A <-> x e. ran ( A i^i _I ) )
14 13 eqriv 
 |-  Fix A = ran ( A i^i _I )