Step |
Hyp |
Ref |
Expression |
1 |
|
fprr.1 |
⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 ) |
2 |
1
|
fpr1 |
⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Fn 𝐴 ) |
3 |
2
|
fndmd |
⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → dom 𝐹 = 𝐴 ) |
4 |
3
|
eleq2d |
⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴 ) ) |
5 |
4
|
biimpar |
⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑋 ∈ 𝐴 ) → 𝑋 ∈ dom 𝐹 ) |
6 |
|
fveq2 |
⊢ ( 𝑦 = 𝑋 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑋 ) ) |
7 |
|
id |
⊢ ( 𝑦 = 𝑋 → 𝑦 = 𝑋 ) |
8 |
|
predeq3 |
⊢ ( 𝑦 = 𝑋 → Pred ( 𝑅 , 𝐴 , 𝑦 ) = Pred ( 𝑅 , 𝐴 , 𝑋 ) ) |
9 |
8
|
reseq2d |
⊢ ( 𝑦 = 𝑋 → ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) |
10 |
7 9
|
oveq12d |
⊢ ( 𝑦 = 𝑋 → ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) |
11 |
6 10
|
eqeq12d |
⊢ ( 𝑦 = 𝑋 → ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑦 = 𝑋 → ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ↔ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) ) ) |
13 |
|
eqid |
⊢ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } = { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } |
14 |
13
|
frrlem1 |
⊢ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } |
15 |
14 1
|
fprlem1 |
⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑔 ∈ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } ∧ ℎ ∈ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } ) ) → ( ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) → 𝑢 = 𝑣 ) ) |
16 |
14 1 15
|
frrlem10 |
⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑦 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) |
17 |
16
|
expcom |
⊢ ( 𝑦 ∈ dom 𝐹 → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) |
18 |
12 17
|
vtoclga |
⊢ ( 𝑋 ∈ dom 𝐹 → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) ) |
19 |
18
|
impcom |
⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑋 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) |
20 |
5 19
|
syldan |
⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) |