Step |
Hyp |
Ref |
Expression |
1 |
|
ss0b |
⊢ ( ( 𝐹 supp 𝑍 ) ⊆ ∅ ↔ ( 𝐹 supp 𝑍 ) = ∅ ) |
2 |
|
un0 |
⊢ ( 𝐴 ∪ ∅ ) = 𝐴 |
3 |
|
uncom |
⊢ ( 𝐴 ∪ ∅ ) = ( ∅ ∪ 𝐴 ) |
4 |
2 3
|
eqtr3i |
⊢ 𝐴 = ( ∅ ∪ 𝐴 ) |
5 |
4
|
fneq2i |
⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 Fn ( ∅ ∪ 𝐴 ) ) |
6 |
5
|
biimpi |
⊢ ( 𝐹 Fn 𝐴 → 𝐹 Fn ( ∅ ∪ 𝐴 ) ) |
7 |
6
|
3ad2ant1 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) → 𝐹 Fn ( ∅ ∪ 𝐴 ) ) |
8 |
|
fnex |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ) → 𝐹 ∈ V ) |
9 |
8
|
3adant3 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) → 𝐹 ∈ V ) |
10 |
|
simp3 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) → 𝑍 ∈ 𝑉 ) |
11 |
|
0in |
⊢ ( ∅ ∩ 𝐴 ) = ∅ |
12 |
11
|
a1i |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) → ( ∅ ∩ 𝐴 ) = ∅ ) |
13 |
|
fnsuppres |
⊢ ( ( 𝐹 Fn ( ∅ ∪ 𝐴 ) ∧ ( 𝐹 ∈ V ∧ 𝑍 ∈ 𝑉 ) ∧ ( ∅ ∩ 𝐴 ) = ∅ ) → ( ( 𝐹 supp 𝑍 ) ⊆ ∅ ↔ ( 𝐹 ↾ 𝐴 ) = ( 𝐴 × { 𝑍 } ) ) ) |
14 |
7 9 10 12 13
|
syl121anc |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) → ( ( 𝐹 supp 𝑍 ) ⊆ ∅ ↔ ( 𝐹 ↾ 𝐴 ) = ( 𝐴 × { 𝑍 } ) ) ) |
15 |
1 14
|
bitr3id |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) → ( ( 𝐹 supp 𝑍 ) = ∅ ↔ ( 𝐹 ↾ 𝐴 ) = ( 𝐴 × { 𝑍 } ) ) ) |
16 |
|
fnresdm |
⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ 𝐴 ) = 𝐹 ) |
17 |
16
|
3ad2ant1 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 ↾ 𝐴 ) = 𝐹 ) |
18 |
17
|
eqeq1d |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) → ( ( 𝐹 ↾ 𝐴 ) = ( 𝐴 × { 𝑍 } ) ↔ 𝐹 = ( 𝐴 × { 𝑍 } ) ) ) |
19 |
15 18
|
bitrd |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) → ( ( 𝐹 supp 𝑍 ) = ∅ ↔ 𝐹 = ( 𝐴 × { 𝑍 } ) ) ) |