Metamath Proof Explorer


Theorem elfunsALTV4

Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021)

Ref Expression
Assertion elfunsALTV4
|- ( F e. FunsALTV <-> ( A. u E* x u F x /\ F e. Rels ) )

Proof

Step Hyp Ref Expression
1 elfunsALTV
 |-  ( F e. FunsALTV <-> ( ,~ F e. CnvRefRels /\ F e. Rels ) )
2 cosselcnvrefrels4
 |-  ( ,~ F e. CnvRefRels <-> ( A. u E* x u F x /\ ,~ F e. Rels ) )
3 cosselrels
 |-  ( F e. Rels -> ,~ F e. Rels )
4 3 biantrud
 |-  ( F e. Rels -> ( A. u E* x u F x <-> ( A. u E* x u F x /\ ,~ F e. Rels ) ) )
5 2 4 bitr4id
 |-  ( F e. Rels -> ( ,~ F e. CnvRefRels <-> A. u E* x u F x ) )
6 5 pm5.32ri
 |-  ( ( ,~ F e. CnvRefRels /\ F e. Rels ) <-> ( A. u E* x u F x /\ F e. Rels ) )
7 1 6 bitri
 |-  ( F e. FunsALTV <-> ( A. u E* x u F x /\ F e. Rels ) )