Metamath Proof Explorer


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 𝐻 = ( ∈ ( 1 -aryF 𝑋 ) ↦ ( 𝑥𝑋 ↦ ( ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) )
Assertion 1arymaptf1o ( 𝑋𝑉𝐻 : ( 1 -aryF 𝑋 ) –1-1-onto→ ( 𝑋m 𝑋 ) )

Proof

Step Hyp Ref Expression
1 1arymaptfv.h 𝐻 = ( ∈ ( 1 -aryF 𝑋 ) ↦ ( 𝑥𝑋 ↦ ( ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) )
2 1 1arymaptf1 ( 𝑋𝑉𝐻 : ( 1 -aryF 𝑋 ) –1-1→ ( 𝑋m 𝑋 ) )
3 1 1arymaptfo ( 𝑋𝑉𝐻 : ( 1 -aryF 𝑋 ) –onto→ ( 𝑋m 𝑋 ) )
4 df-f1o ( 𝐻 : ( 1 -aryF 𝑋 ) –1-1-onto→ ( 𝑋m 𝑋 ) ↔ ( 𝐻 : ( 1 -aryF 𝑋 ) –1-1→ ( 𝑋m 𝑋 ) ∧ 𝐻 : ( 1 -aryF 𝑋 ) –onto→ ( 𝑋m 𝑋 ) ) )
5 2 3 4 sylanbrc ( 𝑋𝑉𝐻 : ( 1 -aryF 𝑋 ) –1-1-onto→ ( 𝑋m 𝑋 ) )