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Metamath Proof Explorer

Theorem elfunsALTV5

Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 5-Sep-2021)

Ref Expression
Assertion elfunsALTV5 
|- ( F e. FunsALTV <-> ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) /\ F e. Rels ) )

Proof

Step Hyp Ref Expression
1 elfunsALTV 
 |-  ( F e. FunsALTV <-> ( ,~ F e. CnvRefRels /\ F e. Rels ) )
2 cosselcnvrefrels5 
 |-  ( ,~ F e. CnvRefRels <-> ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) /\ ,~ F e. Rels ) )
3 cosselrels 
 |-  ( F e. Rels -> ,~ F e. Rels )
4 3 biantrud 
 |-  ( F e. Rels -> ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) <-> ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) /\ ,~ F e. Rels ) ) )
5 2 4 bitr4id 
 |-  ( F e. Rels -> ( ,~ F e. CnvRefRels <-> A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) ) )
6 5 pm5.32ri 
 |-  ( ( ,~ F e. CnvRefRels /\ F e. Rels ) <-> ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) /\ F e. Rels ) )
7 1 6 bitri 
 |-  ( F e. FunsALTV <-> ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) /\ F e. Rels ) )