Metamath Proof Explorer


Theorem fv1arycl

Description: Closure of a unary (endo)function. (Contributed by AV, 18-May-2024)

Ref Expression
Assertion fv1arycl ( ( 𝐺 ∈ ( 1 -aryF 𝑋 ) ∧ 𝐴𝑋 ) → ( 𝐺 ‘ { ⟨ 0 , 𝐴 ⟩ } ) ∈ 𝑋 )

Proof

Step Hyp Ref Expression
1 eqid ( 0 ..^ 1 ) = ( 0 ..^ 1 )
2 1 naryrcl ( 𝐺 ∈ ( 1 -aryF 𝑋 ) → ( 1 ∈ ℕ0𝑋 ∈ V ) )
3 1aryfvalel ( 𝑋 ∈ V → ( 𝐺 ∈ ( 1 -aryF 𝑋 ) ↔ 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋 ) )
4 simp2 ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋𝐴𝑋 ) → 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋 )
5 c0ex 0 ∈ V
6 5 a1i ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋𝐴𝑋 ) → 0 ∈ V )
7 simp3 ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋𝐴𝑋 ) → 𝐴𝑋 )
8 6 7 fsnd ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋𝐴𝑋 ) → { ⟨ 0 , 𝐴 ⟩ } : { 0 } ⟶ 𝑋 )
9 simp1 ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋𝐴𝑋 ) → 𝑋 ∈ V )
10 snex { 0 } ∈ V
11 10 a1i ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋𝐴𝑋 ) → { 0 } ∈ V )
12 9 11 elmapd ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋𝐴𝑋 ) → ( { ⟨ 0 , 𝐴 ⟩ } ∈ ( 𝑋m { 0 } ) ↔ { ⟨ 0 , 𝐴 ⟩ } : { 0 } ⟶ 𝑋 ) )
13 8 12 mpbird ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋𝐴𝑋 ) → { ⟨ 0 , 𝐴 ⟩ } ∈ ( 𝑋m { 0 } ) )
14 4 13 ffvelrnd ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋𝐴𝑋 ) → ( 𝐺 ‘ { ⟨ 0 , 𝐴 ⟩ } ) ∈ 𝑋 )
15 14 3exp ( 𝑋 ∈ V → ( 𝐺 : ( 𝑋m { 0 } ) ⟶ 𝑋 → ( 𝐴𝑋 → ( 𝐺 ‘ { ⟨ 0 , 𝐴 ⟩ } ) ∈ 𝑋 ) ) )
16 3 15 sylbid ( 𝑋 ∈ V → ( 𝐺 ∈ ( 1 -aryF 𝑋 ) → ( 𝐴𝑋 → ( 𝐺 ‘ { ⟨ 0 , 𝐴 ⟩ } ) ∈ 𝑋 ) ) )
17 16 adantl ( ( 1 ∈ ℕ0𝑋 ∈ V ) → ( 𝐺 ∈ ( 1 -aryF 𝑋 ) → ( 𝐴𝑋 → ( 𝐺 ‘ { ⟨ 0 , 𝐴 ⟩ } ) ∈ 𝑋 ) ) )
18 2 17 mpcom ( 𝐺 ∈ ( 1 -aryF 𝑋 ) → ( 𝐴𝑋 → ( 𝐺 ‘ { ⟨ 0 , 𝐴 ⟩ } ) ∈ 𝑋 ) )
19 18 imp ( ( 𝐺 ∈ ( 1 -aryF 𝑋 ) ∧ 𝐴𝑋 ) → ( 𝐺 ‘ { ⟨ 0 , 𝐴 ⟩ } ) ∈ 𝑋 )