Step |
Hyp |
Ref |
Expression |
1 |
|
tfrALT.1 |
⊢ 𝐹 = recs ( 𝐺 ) |
2 |
|
epweon |
⊢ E We On |
3 |
|
epse |
⊢ E Se On |
4 |
|
df-recs |
⊢ recs ( 𝐺 ) = wrecs ( E , On , 𝐺 ) |
5 |
1 4
|
eqtri |
⊢ 𝐹 = wrecs ( E , On , 𝐺 ) |
6 |
5
|
wfr2 |
⊢ ( ( ( E We On ∧ E Se On ) ∧ 𝐴 ∈ On ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( E , On , 𝐴 ) ) ) ) |
7 |
2 3 6
|
mpanl12 |
⊢ ( 𝐴 ∈ On → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( E , On , 𝐴 ) ) ) ) |
8 |
|
predon |
⊢ ( 𝐴 ∈ On → Pred ( E , On , 𝐴 ) = 𝐴 ) |
9 |
8
|
reseq2d |
⊢ ( 𝐴 ∈ On → ( 𝐹 ↾ Pred ( E , On , 𝐴 ) ) = ( 𝐹 ↾ 𝐴 ) ) |
10 |
9
|
fveq2d |
⊢ ( 𝐴 ∈ On → ( 𝐺 ‘ ( 𝐹 ↾ Pred ( E , On , 𝐴 ) ) ) = ( 𝐺 ‘ ( 𝐹 ↾ 𝐴 ) ) ) |
11 |
7 10
|
eqtrd |
⊢ ( 𝐴 ∈ On → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ ( 𝐹 ↾ 𝐴 ) ) ) |