Step |
Hyp |
Ref |
Expression |
1 |
|
dsmmval.b |
⊢ 𝐵 = { 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ∈ Fin } |
2 |
|
elex |
⊢ ( 𝑅 ∈ 𝑉 → 𝑅 ∈ V ) |
3 |
1
|
ssrab3 |
⊢ 𝐵 ⊆ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) |
4 |
|
eqid |
⊢ ( ( 𝑆 Xs 𝑅 ) ↾s 𝐵 ) = ( ( 𝑆 Xs 𝑅 ) ↾s 𝐵 ) |
5 |
|
eqid |
⊢ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) = ( Base ‘ ( 𝑆 Xs 𝑅 ) ) |
6 |
4 5
|
ressbas2 |
⊢ ( 𝐵 ⊆ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) → 𝐵 = ( Base ‘ ( ( 𝑆 Xs 𝑅 ) ↾s 𝐵 ) ) ) |
7 |
3 6
|
ax-mp |
⊢ 𝐵 = ( Base ‘ ( ( 𝑆 Xs 𝑅 ) ↾s 𝐵 ) ) |
8 |
1
|
dsmmval |
⊢ ( 𝑅 ∈ V → ( 𝑆 ⊕m 𝑅 ) = ( ( 𝑆 Xs 𝑅 ) ↾s 𝐵 ) ) |
9 |
8
|
fveq2d |
⊢ ( 𝑅 ∈ V → ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) = ( Base ‘ ( ( 𝑆 Xs 𝑅 ) ↾s 𝐵 ) ) ) |
10 |
7 9
|
eqtr4id |
⊢ ( 𝑅 ∈ V → 𝐵 = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) ) |
11 |
2 10
|
syl |
⊢ ( 𝑅 ∈ 𝑉 → 𝐵 = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) ) |