Step |
Hyp |
Ref |
Expression |
1 |
|
quselbas.e |
⊢ ∼ = ( 𝐺 ~QG 𝑆 ) |
2 |
|
quselbas.u |
⊢ 𝑈 = ( 𝐺 /s ∼ ) |
3 |
|
quselbas.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
4 |
2
|
a1i |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ) → 𝑈 = ( 𝐺 /s ∼ ) ) |
5 |
3
|
a1i |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ) → 𝐵 = ( Base ‘ 𝐺 ) ) |
6 |
1
|
ovexi |
⊢ ∼ ∈ V |
7 |
6
|
a1i |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ) → ∼ ∈ V ) |
8 |
|
simpl |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ) → 𝐺 ∈ 𝑉 ) |
9 |
4 5 7 8
|
qusbas |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ) → ( 𝐵 / ∼ ) = ( Base ‘ 𝑈 ) ) |
10 |
9
|
eqcomd |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ) → ( Base ‘ 𝑈 ) = ( 𝐵 / ∼ ) ) |
11 |
10
|
eleq2d |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ) → ( 𝑋 ∈ ( Base ‘ 𝑈 ) ↔ 𝑋 ∈ ( 𝐵 / ∼ ) ) ) |
12 |
|
elqsg |
⊢ ( 𝑋 ∈ 𝑊 → ( 𝑋 ∈ ( 𝐵 / ∼ ) ↔ ∃ 𝑥 ∈ 𝐵 𝑋 = [ 𝑥 ] ∼ ) ) |
13 |
12
|
adantl |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ) → ( 𝑋 ∈ ( 𝐵 / ∼ ) ↔ ∃ 𝑥 ∈ 𝐵 𝑋 = [ 𝑥 ] ∼ ) ) |
14 |
11 13
|
bitrd |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ) → ( 𝑋 ∈ ( Base ‘ 𝑈 ) ↔ ∃ 𝑥 ∈ 𝐵 𝑋 = [ 𝑥 ] ∼ ) ) |