Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme31.o |
⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) |
2 |
|
cdleme31.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , 𝑂 , 𝑥 ) ) |
3 |
|
cdleme31.c |
⊢ 𝐶 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑧 = ( 𝑁 ∨ ( 𝑋 ∧ 𝑊 ) ) ) ) |
4 |
|
riotaex |
⊢ ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑧 = ( 𝑁 ∨ ( 𝑋 ∧ 𝑊 ) ) ) ) ∈ V |
5 |
3 4
|
eqeltri |
⊢ 𝐶 ∈ V |
6 |
|
ifexg |
⊢ ( ( 𝐶 ∈ V ∧ 𝑋 ∈ 𝐵 ) → if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) , 𝐶 , 𝑋 ) ∈ V ) |
7 |
5 6
|
mpan |
⊢ ( 𝑋 ∈ 𝐵 → if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) , 𝐶 , 𝑋 ) ∈ V ) |
8 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊 ) ) |
9 |
8
|
notbid |
⊢ ( 𝑥 = 𝑋 → ( ¬ 𝑥 ≤ 𝑊 ↔ ¬ 𝑋 ≤ 𝑊 ) ) |
10 |
9
|
anbi2d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) ↔ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) |
11 |
|
oveq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ∧ 𝑊 ) = ( 𝑋 ∧ 𝑊 ) ) |
12 |
11
|
oveq2d |
⊢ ( 𝑥 = 𝑋 → ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) ) |
13 |
|
id |
⊢ ( 𝑥 = 𝑋 → 𝑥 = 𝑋 ) |
14 |
12 13
|
eqeq12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ↔ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) |
15 |
14
|
anbi2d |
⊢ ( 𝑥 = 𝑋 → ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) ↔ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) |
16 |
11
|
oveq2d |
⊢ ( 𝑥 = 𝑋 → ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) = ( 𝑁 ∨ ( 𝑋 ∧ 𝑊 ) ) ) |
17 |
16
|
eqeq2d |
⊢ ( 𝑥 = 𝑋 → ( 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ↔ 𝑧 = ( 𝑁 ∨ ( 𝑋 ∧ 𝑊 ) ) ) ) |
18 |
15 17
|
imbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) ↔ ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑧 = ( 𝑁 ∨ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
19 |
18
|
ralbidv |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) ↔ ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑧 = ( 𝑁 ∨ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
20 |
19
|
riotabidv |
⊢ ( 𝑥 = 𝑋 → ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑧 = ( 𝑁 ∨ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
21 |
20 1 3
|
3eqtr4g |
⊢ ( 𝑥 = 𝑋 → 𝑂 = 𝐶 ) |
22 |
10 21 13
|
ifbieq12d |
⊢ ( 𝑥 = 𝑋 → if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , 𝑂 , 𝑥 ) = if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) , 𝐶 , 𝑋 ) ) |
23 |
22 2
|
fvmptg |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) , 𝐶 , 𝑋 ) ∈ V ) → ( 𝐹 ‘ 𝑋 ) = if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) , 𝐶 , 𝑋 ) ) |
24 |
7 23
|
mpdan |
⊢ ( 𝑋 ∈ 𝐵 → ( 𝐹 ‘ 𝑋 ) = if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) , 𝐶 , 𝑋 ) ) |