Theorem 1arymaptf1o

Description: The mapping of unary (endo)functions is a one-to-one function onto the set of endofunctions. (Contributed by AV, 19-May-2024)

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

Proof

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