Step |
Hyp |
Ref |
Expression |
1 |
|
tpostpos |
⊢ tpos tpos 𝐹 = ( 𝐹 ∩ ( ( ( V × V ) ∪ { ∅ } ) × V ) ) |
2 |
|
relrelss |
⊢ ( ( Rel 𝐹 ∧ Rel dom 𝐹 ) ↔ 𝐹 ⊆ ( ( V × V ) × V ) ) |
3 |
|
ssun1 |
⊢ ( V × V ) ⊆ ( ( V × V ) ∪ { ∅ } ) |
4 |
|
xpss1 |
⊢ ( ( V × V ) ⊆ ( ( V × V ) ∪ { ∅ } ) → ( ( V × V ) × V ) ⊆ ( ( ( V × V ) ∪ { ∅ } ) × V ) ) |
5 |
3 4
|
ax-mp |
⊢ ( ( V × V ) × V ) ⊆ ( ( ( V × V ) ∪ { ∅ } ) × V ) |
6 |
|
sstr |
⊢ ( ( 𝐹 ⊆ ( ( V × V ) × V ) ∧ ( ( V × V ) × V ) ⊆ ( ( ( V × V ) ∪ { ∅ } ) × V ) ) → 𝐹 ⊆ ( ( ( V × V ) ∪ { ∅ } ) × V ) ) |
7 |
5 6
|
mpan2 |
⊢ ( 𝐹 ⊆ ( ( V × V ) × V ) → 𝐹 ⊆ ( ( ( V × V ) ∪ { ∅ } ) × V ) ) |
8 |
2 7
|
sylbi |
⊢ ( ( Rel 𝐹 ∧ Rel dom 𝐹 ) → 𝐹 ⊆ ( ( ( V × V ) ∪ { ∅ } ) × V ) ) |
9 |
|
df-ss |
⊢ ( 𝐹 ⊆ ( ( ( V × V ) ∪ { ∅ } ) × V ) ↔ ( 𝐹 ∩ ( ( ( V × V ) ∪ { ∅ } ) × V ) ) = 𝐹 ) |
10 |
8 9
|
sylib |
⊢ ( ( Rel 𝐹 ∧ Rel dom 𝐹 ) → ( 𝐹 ∩ ( ( ( V × V ) ∪ { ∅ } ) × V ) ) = 𝐹 ) |
11 |
1 10
|
eqtrid |
⊢ ( ( Rel 𝐹 ∧ Rel dom 𝐹 ) → tpos tpos 𝐹 = 𝐹 ) |