Metamath Proof Explorer


Theorem 0aryfvalel

Description: A nullary (endo)function on a set X is a singleton of an ordered pair with the empty set as first component. A nullary function represents a constant: ( F(/) ) = C with C e. X , see also 0aryfvalelfv . Instead of ( F(/) ) , nullary functions are usually written as F ( ) in literature. (Contributed by AV, 15-May-2024)

Ref Expression
Assertion 0aryfvalel ( 𝑋𝑉 → ( 𝐹 ∈ ( 0 -aryF 𝑋 ) ↔ ∃ 𝑥𝑋 𝐹 = { ⟨ ∅ , 𝑥 ⟩ } ) )

Proof

Step Hyp Ref Expression
1 0nn0 0 ∈ ℕ0
2 fzo0 ( 0 ..^ 0 ) = ∅
3 2 eqcomi ∅ = ( 0 ..^ 0 )
4 3 naryfvalel ( ( 0 ∈ ℕ0𝑋𝑉 ) → ( 𝐹 ∈ ( 0 -aryF 𝑋 ) ↔ 𝐹 : ( 𝑋m ∅ ) ⟶ 𝑋 ) )
5 1 4 mpan ( 𝑋𝑉 → ( 𝐹 ∈ ( 0 -aryF 𝑋 ) ↔ 𝐹 : ( 𝑋m ∅ ) ⟶ 𝑋 ) )
6 mapdm0 ( 𝑋𝑉 → ( 𝑋m ∅ ) = { ∅ } )
7 6 feq2d ( 𝑋𝑉 → ( 𝐹 : ( 𝑋m ∅ ) ⟶ 𝑋𝐹 : { ∅ } ⟶ 𝑋 ) )
8 0ex ∅ ∈ V
9 8 fsn2 ( 𝐹 : { ∅ } ⟶ 𝑋 ↔ ( ( 𝐹 ‘ ∅ ) ∈ 𝑋𝐹 = { ⟨ ∅ , ( 𝐹 ‘ ∅ ) ⟩ } ) )
10 opeq2 ( 𝑥 = ( 𝐹 ‘ ∅ ) → ⟨ ∅ , 𝑥 ⟩ = ⟨ ∅ , ( 𝐹 ‘ ∅ ) ⟩ )
11 10 sneqd ( 𝑥 = ( 𝐹 ‘ ∅ ) → { ⟨ ∅ , 𝑥 ⟩ } = { ⟨ ∅ , ( 𝐹 ‘ ∅ ) ⟩ } )
12 11 rspceeqv ( ( ( 𝐹 ‘ ∅ ) ∈ 𝑋𝐹 = { ⟨ ∅ , ( 𝐹 ‘ ∅ ) ⟩ } ) → ∃ 𝑥𝑋 𝐹 = { ⟨ ∅ , 𝑥 ⟩ } )
13 9 12 sylbi ( 𝐹 : { ∅ } ⟶ 𝑋 → ∃ 𝑥𝑋 𝐹 = { ⟨ ∅ , 𝑥 ⟩ } )
14 8 a1i ( 𝑥𝑋 → ∅ ∈ V )
15 id ( 𝑥𝑋𝑥𝑋 )
16 14 15 fsnd ( 𝑥𝑋 → { ⟨ ∅ , 𝑥 ⟩ } : { ∅ } ⟶ 𝑋 )
17 feq1 ( 𝐹 = { ⟨ ∅ , 𝑥 ⟩ } → ( 𝐹 : { ∅ } ⟶ 𝑋 ↔ { ⟨ ∅ , 𝑥 ⟩ } : { ∅ } ⟶ 𝑋 ) )
18 16 17 syl5ibrcom ( 𝑥𝑋 → ( 𝐹 = { ⟨ ∅ , 𝑥 ⟩ } → 𝐹 : { ∅ } ⟶ 𝑋 ) )
19 18 rexlimiv ( ∃ 𝑥𝑋 𝐹 = { ⟨ ∅ , 𝑥 ⟩ } → 𝐹 : { ∅ } ⟶ 𝑋 )
20 13 19 impbii ( 𝐹 : { ∅ } ⟶ 𝑋 ↔ ∃ 𝑥𝑋 𝐹 = { ⟨ ∅ , 𝑥 ⟩ } )
21 20 a1i ( 𝑋𝑉 → ( 𝐹 : { ∅ } ⟶ 𝑋 ↔ ∃ 𝑥𝑋 𝐹 = { ⟨ ∅ , 𝑥 ⟩ } ) )
22 5 7 21 3bitrd ( 𝑋𝑉 → ( 𝐹 ∈ ( 0 -aryF 𝑋 ) ↔ ∃ 𝑥𝑋 𝐹 = { ⟨ ∅ , 𝑥 ⟩ } ) )