Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( 0 ..^ 1 ) = ( 0 ..^ 1 ) |
2 |
1
|
naryrcl |
⊢ ( 𝐺 ∈ ( 1 -aryF 𝑋 ) → ( 1 ∈ ℕ0 ∧ 𝑋 ∈ V ) ) |
3 |
|
1aryfvalel |
⊢ ( 𝑋 ∈ V → ( 𝐺 ∈ ( 1 -aryF 𝑋 ) ↔ 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ) ) |
4 |
|
simp2 |
⊢ ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ) |
5 |
|
c0ex |
⊢ 0 ∈ V |
6 |
5
|
a1i |
⊢ ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → 0 ∈ V ) |
7 |
|
simp3 |
⊢ ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 ) |
8 |
6 7
|
fsnd |
⊢ ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → { 〈 0 , 𝐴 〉 } : { 0 } ⟶ 𝑋 ) |
9 |
|
simp1 |
⊢ ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → 𝑋 ∈ V ) |
10 |
|
snex |
⊢ { 0 } ∈ V |
11 |
10
|
a1i |
⊢ ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → { 0 } ∈ V ) |
12 |
9 11
|
elmapd |
⊢ ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( { 〈 0 , 𝐴 〉 } ∈ ( 𝑋 ↑m { 0 } ) ↔ { 〈 0 , 𝐴 〉 } : { 0 } ⟶ 𝑋 ) ) |
13 |
8 12
|
mpbird |
⊢ ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → { 〈 0 , 𝐴 〉 } ∈ ( 𝑋 ↑m { 0 } ) ) |
14 |
4 13
|
ffvelrnd |
⊢ ( ( 𝑋 ∈ V ∧ 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐺 ‘ { 〈 0 , 𝐴 〉 } ) ∈ 𝑋 ) |
15 |
14
|
3exp |
⊢ ( 𝑋 ∈ V → ( 𝐺 : ( 𝑋 ↑m { 0 } ) ⟶ 𝑋 → ( 𝐴 ∈ 𝑋 → ( 𝐺 ‘ { 〈 0 , 𝐴 〉 } ) ∈ 𝑋 ) ) ) |
16 |
3 15
|
sylbid |
⊢ ( 𝑋 ∈ V → ( 𝐺 ∈ ( 1 -aryF 𝑋 ) → ( 𝐴 ∈ 𝑋 → ( 𝐺 ‘ { 〈 0 , 𝐴 〉 } ) ∈ 𝑋 ) ) ) |
17 |
16
|
adantl |
⊢ ( ( 1 ∈ ℕ0 ∧ 𝑋 ∈ V ) → ( 𝐺 ∈ ( 1 -aryF 𝑋 ) → ( 𝐴 ∈ 𝑋 → ( 𝐺 ‘ { 〈 0 , 𝐴 〉 } ) ∈ 𝑋 ) ) ) |
18 |
2 17
|
mpcom |
⊢ ( 𝐺 ∈ ( 1 -aryF 𝑋 ) → ( 𝐴 ∈ 𝑋 → ( 𝐺 ‘ { 〈 0 , 𝐴 〉 } ) ∈ 𝑋 ) ) |
19 |
18
|
imp |
⊢ ( ( 𝐺 ∈ ( 1 -aryF 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐺 ‘ { 〈 0 , 𝐴 〉 } ) ∈ 𝑋 ) |