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 = ( ℎ ∈ ( 1 -aryF 𝑋 ) ↦ ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) )
Assertion	1arymaptf1o	⊢ ( 𝑋 ∈ 𝑉 → 𝐻 : ( 1 -aryF 𝑋 ) –1-1-onto→ ( 𝑋 ↑m 𝑋 ) )

Proof

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