Step |
Hyp |
Ref |
Expression |
1 |
|
csbin |
⊢ ⦋ 𝐴 / 𝑥 ⦌ ( 𝐷 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐷 ∩ ⦋ 𝐴 / 𝑥 ⦌ ( ◡ 𝑅 “ { 𝑋 } ) ) |
2 |
|
csbima12 |
⊢ ⦋ 𝐴 / 𝑥 ⦌ ( ◡ 𝑅 “ { 𝑋 } ) = ( ⦋ 𝐴 / 𝑥 ⦌ ◡ 𝑅 “ ⦋ 𝐴 / 𝑥 ⦌ { 𝑋 } ) |
3 |
|
csbcnv |
⊢ ◡ ⦋ 𝐴 / 𝑥 ⦌ 𝑅 = ⦋ 𝐴 / 𝑥 ⦌ ◡ 𝑅 |
4 |
3
|
imaeq1i |
⊢ ( ◡ ⦋ 𝐴 / 𝑥 ⦌ 𝑅 “ ⦋ 𝐴 / 𝑥 ⦌ { 𝑋 } ) = ( ⦋ 𝐴 / 𝑥 ⦌ ◡ 𝑅 “ ⦋ 𝐴 / 𝑥 ⦌ { 𝑋 } ) |
5 |
|
csbsng |
⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ { 𝑋 } = { ⦋ 𝐴 / 𝑥 ⦌ 𝑋 } ) |
6 |
5
|
imaeq2d |
⊢ ( 𝐴 ∈ 𝑉 → ( ◡ ⦋ 𝐴 / 𝑥 ⦌ 𝑅 “ ⦋ 𝐴 / 𝑥 ⦌ { 𝑋 } ) = ( ◡ ⦋ 𝐴 / 𝑥 ⦌ 𝑅 “ { ⦋ 𝐴 / 𝑥 ⦌ 𝑋 } ) ) |
7 |
4 6
|
eqtr3id |
⊢ ( 𝐴 ∈ 𝑉 → ( ⦋ 𝐴 / 𝑥 ⦌ ◡ 𝑅 “ ⦋ 𝐴 / 𝑥 ⦌ { 𝑋 } ) = ( ◡ ⦋ 𝐴 / 𝑥 ⦌ 𝑅 “ { ⦋ 𝐴 / 𝑥 ⦌ 𝑋 } ) ) |
8 |
2 7
|
eqtrid |
⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( ◡ 𝑅 “ { 𝑋 } ) = ( ◡ ⦋ 𝐴 / 𝑥 ⦌ 𝑅 “ { ⦋ 𝐴 / 𝑥 ⦌ 𝑋 } ) ) |
9 |
8
|
ineq2d |
⊢ ( 𝐴 ∈ 𝑉 → ( ⦋ 𝐴 / 𝑥 ⦌ 𝐷 ∩ ⦋ 𝐴 / 𝑥 ⦌ ( ◡ 𝑅 “ { 𝑋 } ) ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐷 ∩ ( ◡ ⦋ 𝐴 / 𝑥 ⦌ 𝑅 “ { ⦋ 𝐴 / 𝑥 ⦌ 𝑋 } ) ) ) |
10 |
1 9
|
eqtrid |
⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐷 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐷 ∩ ( ◡ ⦋ 𝐴 / 𝑥 ⦌ 𝑅 “ { ⦋ 𝐴 / 𝑥 ⦌ 𝑋 } ) ) ) |
11 |
|
df-pred |
⊢ Pred ( 𝑅 , 𝐷 , 𝑋 ) = ( 𝐷 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) |
12 |
11
|
csbeq2i |
⊢ ⦋ 𝐴 / 𝑥 ⦌ Pred ( 𝑅 , 𝐷 , 𝑋 ) = ⦋ 𝐴 / 𝑥 ⦌ ( 𝐷 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) |
13 |
|
df-pred |
⊢ Pred ( ⦋ 𝐴 / 𝑥 ⦌ 𝑅 , ⦋ 𝐴 / 𝑥 ⦌ 𝐷 , ⦋ 𝐴 / 𝑥 ⦌ 𝑋 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐷 ∩ ( ◡ ⦋ 𝐴 / 𝑥 ⦌ 𝑅 “ { ⦋ 𝐴 / 𝑥 ⦌ 𝑋 } ) ) |
14 |
10 12 13
|
3eqtr4g |
⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ Pred ( 𝑅 , 𝐷 , 𝑋 ) = Pred ( ⦋ 𝐴 / 𝑥 ⦌ 𝑅 , ⦋ 𝐴 / 𝑥 ⦌ 𝐷 , ⦋ 𝐴 / 𝑥 ⦌ 𝑋 ) ) |