Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1015.1 |
⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
2 |
|
bnj1015.2 |
⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
3 |
|
bnj1015.13 |
⊢ 𝐷 = ( ω ∖ { ∅ } ) |
4 |
|
bnj1015.14 |
⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) } |
5 |
|
bnj1015.15 |
⊢ 𝐺 ∈ 𝑉 |
6 |
|
bnj1015.16 |
⊢ 𝐽 ∈ 𝑉 |
7 |
6
|
elexi |
⊢ 𝐽 ∈ V |
8 |
|
eleq1 |
⊢ ( 𝑗 = 𝐽 → ( 𝑗 ∈ dom 𝐺 ↔ 𝐽 ∈ dom 𝐺 ) ) |
9 |
8
|
anbi2d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺 ) ↔ ( 𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺 ) ) ) |
10 |
|
fveq2 |
⊢ ( 𝑗 = 𝐽 → ( 𝐺 ‘ 𝑗 ) = ( 𝐺 ‘ 𝐽 ) ) |
11 |
10
|
sseq1d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝐺 ‘ 𝑗 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝐺 ‘ 𝐽 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
12 |
9 11
|
imbi12d |
⊢ ( 𝑗 = 𝐽 → ( ( ( 𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺 ) → ( 𝐺 ‘ 𝑗 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ↔ ( ( 𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺 ) → ( 𝐺 ‘ 𝐽 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) ) |
13 |
5
|
elexi |
⊢ 𝐺 ∈ V |
14 |
|
eleq1 |
⊢ ( 𝑔 = 𝐺 → ( 𝑔 ∈ 𝐵 ↔ 𝐺 ∈ 𝐵 ) ) |
15 |
|
dmeq |
⊢ ( 𝑔 = 𝐺 → dom 𝑔 = dom 𝐺 ) |
16 |
15
|
eleq2d |
⊢ ( 𝑔 = 𝐺 → ( 𝑗 ∈ dom 𝑔 ↔ 𝑗 ∈ dom 𝐺 ) ) |
17 |
14 16
|
anbi12d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔 ) ↔ ( 𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺 ) ) ) |
18 |
|
fveq1 |
⊢ ( 𝑔 = 𝐺 → ( 𝑔 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) |
19 |
18
|
sseq1d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝑔 ‘ 𝑗 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝐺 ‘ 𝑗 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
20 |
17 19
|
imbi12d |
⊢ ( 𝑔 = 𝐺 → ( ( ( 𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔 ) → ( 𝑔 ‘ 𝑗 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ↔ ( ( 𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺 ) → ( 𝐺 ‘ 𝑗 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) ) |
21 |
1 2 3 4
|
bnj1014 |
⊢ ( ( 𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔 ) → ( 𝑔 ‘ 𝑗 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) |
22 |
13 20 21
|
vtocl |
⊢ ( ( 𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺 ) → ( 𝐺 ‘ 𝑗 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) |
23 |
7 12 22
|
vtocl |
⊢ ( ( 𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺 ) → ( 𝐺 ‘ 𝐽 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) |