Step |
Hyp |
Ref |
Expression |
1 |
|
fnwe2.su |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑥 ) → 𝑆 = 𝑈 ) |
2 |
|
fnwe2.t |
⊢ 𝑇 = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ∨ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 𝑈 𝑦 ) ) } |
3 |
|
vex |
⊢ 𝑎 ∈ V |
4 |
|
vex |
⊢ 𝑏 ∈ V |
5 |
|
fveq2 |
⊢ ( 𝑥 = 𝑎 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑎 ) ) |
6 |
|
fveq2 |
⊢ ( 𝑦 = 𝑏 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑏 ) ) |
7 |
5 6
|
breqan12d |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑎 ) 𝑅 ( 𝐹 ‘ 𝑏 ) ) ) |
8 |
5 6
|
eqeqan12d |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) ) ) |
9 |
|
simpl |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → 𝑥 = 𝑎 ) |
10 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑥 ) ∈ V |
11 |
10 1
|
csbie |
⊢ ⦋ ( 𝐹 ‘ 𝑥 ) / 𝑧 ⦌ 𝑆 = 𝑈 |
12 |
5
|
csbeq1d |
⊢ ( 𝑥 = 𝑎 → ⦋ ( 𝐹 ‘ 𝑥 ) / 𝑧 ⦌ 𝑆 = ⦋ ( 𝐹 ‘ 𝑎 ) / 𝑧 ⦌ 𝑆 ) |
13 |
11 12
|
eqtr3id |
⊢ ( 𝑥 = 𝑎 → 𝑈 = ⦋ ( 𝐹 ‘ 𝑎 ) / 𝑧 ⦌ 𝑆 ) |
14 |
13
|
adantr |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → 𝑈 = ⦋ ( 𝐹 ‘ 𝑎 ) / 𝑧 ⦌ 𝑆 ) |
15 |
|
simpr |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → 𝑦 = 𝑏 ) |
16 |
9 14 15
|
breq123d |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → ( 𝑥 𝑈 𝑦 ↔ 𝑎 ⦋ ( 𝐹 ‘ 𝑎 ) / 𝑧 ⦌ 𝑆 𝑏 ) ) |
17 |
8 16
|
anbi12d |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → ( ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 𝑈 𝑦 ) ↔ ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) ∧ 𝑎 ⦋ ( 𝐹 ‘ 𝑎 ) / 𝑧 ⦌ 𝑆 𝑏 ) ) ) |
18 |
7 17
|
orbi12d |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → ( ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ∨ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 𝑈 𝑦 ) ) ↔ ( ( 𝐹 ‘ 𝑎 ) 𝑅 ( 𝐹 ‘ 𝑏 ) ∨ ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) ∧ 𝑎 ⦋ ( 𝐹 ‘ 𝑎 ) / 𝑧 ⦌ 𝑆 𝑏 ) ) ) ) |
19 |
3 4 18 2
|
braba |
⊢ ( 𝑎 𝑇 𝑏 ↔ ( ( 𝐹 ‘ 𝑎 ) 𝑅 ( 𝐹 ‘ 𝑏 ) ∨ ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) ∧ 𝑎 ⦋ ( 𝐹 ‘ 𝑎 ) / 𝑧 ⦌ 𝑆 𝑏 ) ) ) |