Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1121.1 |
⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) |
2 |
|
bnj1121.2 |
⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) ) |
3 |
|
bnj1121.3 |
⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) |
4 |
|
bnj1121.4 |
⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) |
5 |
|
bnj1121.5 |
⊢ ( 𝜂 ↔ ( ( 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ) |
6 |
|
bnj1121.6 |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ∀ 𝑖 ∈ 𝑛 𝜂 ) |
7 |
|
bnj1121.7 |
⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) } |
8 |
|
19.8a |
⊢ ( 𝜒 → ∃ 𝑛 𝜒 ) |
9 |
8
|
bnj707 |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ∃ 𝑛 𝜒 ) |
10 |
3 7
|
bnj1083 |
⊢ ( 𝑓 ∈ 𝐾 ↔ ∃ 𝑛 𝜒 ) |
11 |
9 10
|
sylibr |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑓 ∈ 𝐾 ) |
12 |
4
|
simplbi |
⊢ ( 𝜁 → 𝑖 ∈ 𝑛 ) |
13 |
12
|
bnj708 |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑖 ∈ 𝑛 ) |
14 |
3
|
bnj1235 |
⊢ ( 𝜒 → 𝑓 Fn 𝑛 ) |
15 |
14
|
bnj707 |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑓 Fn 𝑛 ) |
16 |
15
|
fndmd |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → dom 𝑓 = 𝑛 ) |
17 |
13 16
|
eleqtrrd |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑖 ∈ dom 𝑓 ) |
18 |
6 13
|
bnj1294 |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝜂 ) |
19 |
18 5
|
sylib |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ( ( 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ) |
20 |
11 17 19
|
mp2and |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) |
21 |
4
|
simprbi |
⊢ ( 𝜁 → 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) |
22 |
21
|
bnj708 |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) |
23 |
20 22
|
sseldd |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) |