Step |
Hyp |
Ref |
Expression |
1 |
|
funres |
⊢ ( Fun 𝐴 → Fun ( 𝐴 ↾ 𝐵 ) ) |
2 |
1
|
3ad2ant1 |
⊢ ( ( Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴 ) → Fun ( 𝐴 ↾ 𝐵 ) ) |
3 |
|
finresfin |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 ↾ 𝐵 ) ∈ Fin ) |
4 |
3
|
3ad2ant2 |
⊢ ( ( Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴 ) → ( 𝐴 ↾ 𝐵 ) ∈ Fin ) |
5 |
|
hashfun |
⊢ ( ( 𝐴 ↾ 𝐵 ) ∈ Fin → ( Fun ( 𝐴 ↾ 𝐵 ) ↔ ( ♯ ‘ ( 𝐴 ↾ 𝐵 ) ) = ( ♯ ‘ dom ( 𝐴 ↾ 𝐵 ) ) ) ) |
6 |
4 5
|
syl |
⊢ ( ( Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴 ) → ( Fun ( 𝐴 ↾ 𝐵 ) ↔ ( ♯ ‘ ( 𝐴 ↾ 𝐵 ) ) = ( ♯ ‘ dom ( 𝐴 ↾ 𝐵 ) ) ) ) |
7 |
2 6
|
mpbid |
⊢ ( ( Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴 ) → ( ♯ ‘ ( 𝐴 ↾ 𝐵 ) ) = ( ♯ ‘ dom ( 𝐴 ↾ 𝐵 ) ) ) |
8 |
|
ssdmres |
⊢ ( 𝐵 ⊆ dom 𝐴 ↔ dom ( 𝐴 ↾ 𝐵 ) = 𝐵 ) |
9 |
8
|
biimpi |
⊢ ( 𝐵 ⊆ dom 𝐴 → dom ( 𝐴 ↾ 𝐵 ) = 𝐵 ) |
10 |
9
|
3ad2ant3 |
⊢ ( ( Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴 ) → dom ( 𝐴 ↾ 𝐵 ) = 𝐵 ) |
11 |
10
|
fveq2d |
⊢ ( ( Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴 ) → ( ♯ ‘ dom ( 𝐴 ↾ 𝐵 ) ) = ( ♯ ‘ 𝐵 ) ) |
12 |
7 11
|
eqtrd |
⊢ ( ( Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴 ) → ( ♯ ‘ ( 𝐴 ↾ 𝐵 ) ) = ( ♯ ‘ 𝐵 ) ) |