Step |
Hyp |
Ref |
Expression |
1 |
|
rdgsucmptf.1 |
⊢ Ⅎ 𝑥 𝐴 |
2 |
|
rdgsucmptf.2 |
⊢ Ⅎ 𝑥 𝐵 |
3 |
|
rdgsucmptf.3 |
⊢ Ⅎ 𝑥 𝐷 |
4 |
|
rdgsucmptf.4 |
⊢ 𝐹 = rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) |
5 |
|
rdgsucmptf.5 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝐵 ) → 𝐶 = 𝐷 ) |
6 |
4
|
fveq1i |
⊢ ( 𝐹 ‘ suc 𝐵 ) = ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ suc 𝐵 ) |
7 |
|
rdgdmlim |
⊢ Lim dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) |
8 |
|
limsuc |
⊢ ( Lim dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) → ( 𝐵 ∈ dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ↔ suc 𝐵 ∈ dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ) ) |
9 |
7 8
|
ax-mp |
⊢ ( 𝐵 ∈ dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ↔ suc 𝐵 ∈ dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ) |
10 |
|
rdgsucg |
⊢ ( 𝐵 ∈ dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) → ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ suc 𝐵 ) = ( ( 𝑥 ∈ V ↦ 𝐶 ) ‘ ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ 𝐵 ) ) ) |
11 |
4
|
fveq1i |
⊢ ( 𝐹 ‘ 𝐵 ) = ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ 𝐵 ) |
12 |
11
|
fveq2i |
⊢ ( ( 𝑥 ∈ V ↦ 𝐶 ) ‘ ( 𝐹 ‘ 𝐵 ) ) = ( ( 𝑥 ∈ V ↦ 𝐶 ) ‘ ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ 𝐵 ) ) |
13 |
10 12
|
eqtr4di |
⊢ ( 𝐵 ∈ dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) → ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ suc 𝐵 ) = ( ( 𝑥 ∈ V ↦ 𝐶 ) ‘ ( 𝐹 ‘ 𝐵 ) ) ) |
14 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ V ↦ 𝐶 ) |
15 |
14 1
|
nfrdg |
⊢ Ⅎ 𝑥 rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) |
16 |
4 15
|
nfcxfr |
⊢ Ⅎ 𝑥 𝐹 |
17 |
16 2
|
nffv |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝐵 ) |
18 |
|
eqid |
⊢ ( 𝑥 ∈ V ↦ 𝐶 ) = ( 𝑥 ∈ V ↦ 𝐶 ) |
19 |
17 3 5 18
|
fvmptnf |
⊢ ( ¬ 𝐷 ∈ V → ( ( 𝑥 ∈ V ↦ 𝐶 ) ‘ ( 𝐹 ‘ 𝐵 ) ) = ∅ ) |
20 |
13 19
|
sylan9eqr |
⊢ ( ( ¬ 𝐷 ∈ V ∧ 𝐵 ∈ dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ) → ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ suc 𝐵 ) = ∅ ) |
21 |
20
|
ex |
⊢ ( ¬ 𝐷 ∈ V → ( 𝐵 ∈ dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) → ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ suc 𝐵 ) = ∅ ) ) |
22 |
9 21
|
syl5bir |
⊢ ( ¬ 𝐷 ∈ V → ( suc 𝐵 ∈ dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) → ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ suc 𝐵 ) = ∅ ) ) |
23 |
|
ndmfv |
⊢ ( ¬ suc 𝐵 ∈ dom rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) → ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ suc 𝐵 ) = ∅ ) |
24 |
22 23
|
pm2.61d1 |
⊢ ( ¬ 𝐷 ∈ V → ( rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) ‘ suc 𝐵 ) = ∅ ) |
25 |
6 24
|
eqtrid |
⊢ ( ¬ 𝐷 ∈ V → ( 𝐹 ‘ suc 𝐵 ) = ∅ ) |