Theorem 2arymaptf1o

Description: The mapping of binary (endo)functions is a one-to-one function onto the set of binary operations. (Contributed by AV, 23-May-2024)

	Ref	Expression
Hypothesis	2arymaptf.h	No typesetting found for |- H = ( h e. ( 2 -aryF X ) |-> ( x e. X , y e. X |-> ( h ` { <. 0 , x >. , <. 1 , y >. } ) ) ) with typecode |-
Assertion	2arymaptf1o	Could not format assertion : No typesetting found for |- ( X e. V -> H : ( 2 -aryF X ) -1-1-onto-> ( X ^m ( X X. X ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		2arymaptf.h	Could not format H = ( h e. ( 2 -aryF X ) |-> ( x e. X , y e. X |-> ( h ` { <. 0 , x >. , <. 1 , y >. } ) ) ) : No typesetting found for |- H = ( h e. ( 2 -aryF X ) |-> ( x e. X , y e. X |-> ( h ` { <. 0 , x >. , <. 1 , y >. } ) ) ) with typecode |-
2	1	2arymaptf1	Could not format ( X e. V -> H : ( 2 -aryF X ) -1-1-> ( X ^m ( X X. X ) ) ) : No typesetting found for |- ( X e. V -> H : ( 2 -aryF X ) -1-1-> ( X ^m ( X X. X ) ) ) with typecode |-
3	1	2arymaptfo	Could not format ( X e. V -> H : ( 2 -aryF X ) -onto-> ( X ^m ( X X. X ) ) ) : No typesetting found for |- ( X e. V -> H : ( 2 -aryF X ) -onto-> ( X ^m ( X X. X ) ) ) with typecode |-
4		df-f1o	Could not format ( H : ( 2 -aryF X ) -1-1-onto-> ( X ^m ( X X. X ) ) <-> ( H : ( 2 -aryF X ) -1-1-> ( X ^m ( X X. X ) ) /\ H : ( 2 -aryF X ) -onto-> ( X ^m ( X X. X ) ) ) ) : No typesetting found for |- ( H : ( 2 -aryF X ) -1-1-onto-> ( X ^m ( X X. X ) ) <-> ( H : ( 2 -aryF X ) -1-1-> ( X ^m ( X X. X ) ) /\ H : ( 2 -aryF X ) -onto-> ( X ^m ( X X. X ) ) ) ) with typecode |-
5	2 3 4	sylanbrc	Could not format ( X e. V -> H : ( 2 -aryF X ) -1-1-onto-> ( X ^m ( X X. X ) ) ) : No typesetting found for |- ( X e. V -> H : ( 2 -aryF X ) -1-1-onto-> ( X ^m ( X X. X ) ) ) with typecode |-