This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dffunsALTV3

Description: Alternate definition of the class of functions. For the X axis and the Y axis you can convert the right side to { f e. Rels | A. x1 A. y1 A. y2 ( ( x1 f y1 /\ x1 f y2 ) -> y1 = y2 ) } . (Contributed by Peter Mazsa, 30-Aug-2021)

Ref Expression
Assertion dffunsALTV3 
|- FunsALTV = { f e. Rels | A. u A. x A. y ( ( u f x /\ u f y ) -> x = y ) }

Proof

Step Hyp Ref Expression
1 dffunsALTV 
 |-  FunsALTV = { f e. Rels | ,~ f e. CnvRefRels }
2 cosselcnvrefrels3 
 |-  ( ,~ f e. CnvRefRels <-> ( A. u A. x A. y ( ( u f x /\ u f y ) -> x = y ) /\ ,~ f e. Rels ) )
3 cosselrels 
 |-  ( f e. Rels -> ,~ f e. Rels )
4 3 biantrud 
 |-  ( f e. Rels -> ( A. u A. x A. y ( ( u f x /\ u f y ) -> x = y ) <-> ( A. u A. x A. y ( ( u f x /\ u f y ) -> x = y ) /\ ,~ f e. Rels ) ) )
5 2 4 bitr4id 
 |-  ( f e. Rels -> ( ,~ f e. CnvRefRels <-> A. u A. x A. y ( ( u f x /\ u f y ) -> x = y ) ) )
6 1 5 rabimbieq 
 |-  FunsALTV = { f e. Rels | A. u A. x A. y ( ( u f x /\ u f y ) -> x = y ) }