Step |
Hyp |
Ref |
Expression |
1 |
|
cantnfs.s |
⊢ 𝑆 = dom ( 𝐴 CNF 𝐵 ) |
2 |
|
cantnfs.a |
⊢ ( 𝜑 → 𝐴 ∈ On ) |
3 |
|
cantnfs.b |
⊢ ( 𝜑 → 𝐵 ∈ On ) |
4 |
|
cantnflt2.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑆 ) |
5 |
|
cantnflt2.a |
⊢ ( 𝜑 → ∅ ∈ 𝐴 ) |
6 |
|
cantnflt2.c |
⊢ ( 𝜑 → 𝐶 ∈ On ) |
7 |
|
cantnflt2.s |
⊢ ( 𝜑 → ( 𝐹 supp ∅ ) ⊆ 𝐶 ) |
8 |
|
eqid |
⊢ OrdIso ( E , ( 𝐹 supp ∅ ) ) = OrdIso ( E , ( 𝐹 supp ∅ ) ) |
9 |
|
eqid |
⊢ seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝐴 ↑o ( OrdIso ( E , ( 𝐹 supp ∅ ) ) ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( OrdIso ( E , ( 𝐹 supp ∅ ) ) ‘ 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) = seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝐴 ↑o ( OrdIso ( E , ( 𝐹 supp ∅ ) ) ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( OrdIso ( E , ( 𝐹 supp ∅ ) ) ‘ 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) |
10 |
1 2 3 8 4 9
|
cantnfval |
⊢ ( 𝜑 → ( ( 𝐴 CNF 𝐵 ) ‘ 𝐹 ) = ( seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝐴 ↑o ( OrdIso ( E , ( 𝐹 supp ∅ ) ) ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( OrdIso ( E , ( 𝐹 supp ∅ ) ) ‘ 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) ‘ dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ) ) |
11 |
|
ovexd |
⊢ ( 𝜑 → ( 𝐹 supp ∅ ) ∈ V ) |
12 |
8
|
oion |
⊢ ( ( 𝐹 supp ∅ ) ∈ V → dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ∈ On ) |
13 |
|
sucidg |
⊢ ( dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ∈ On → dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ∈ suc dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ) |
14 |
11 12 13
|
3syl |
⊢ ( 𝜑 → dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ∈ suc dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ) |
15 |
1 2 3 8 4
|
cantnfcl |
⊢ ( 𝜑 → ( E We ( 𝐹 supp ∅ ) ∧ dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ∈ ω ) ) |
16 |
15
|
simpld |
⊢ ( 𝜑 → E We ( 𝐹 supp ∅ ) ) |
17 |
8
|
oiiso |
⊢ ( ( ( 𝐹 supp ∅ ) ∈ V ∧ E We ( 𝐹 supp ∅ ) ) → OrdIso ( E , ( 𝐹 supp ∅ ) ) Isom E , E ( dom OrdIso ( E , ( 𝐹 supp ∅ ) ) , ( 𝐹 supp ∅ ) ) ) |
18 |
11 16 17
|
syl2anc |
⊢ ( 𝜑 → OrdIso ( E , ( 𝐹 supp ∅ ) ) Isom E , E ( dom OrdIso ( E , ( 𝐹 supp ∅ ) ) , ( 𝐹 supp ∅ ) ) ) |
19 |
|
isof1o |
⊢ ( OrdIso ( E , ( 𝐹 supp ∅ ) ) Isom E , E ( dom OrdIso ( E , ( 𝐹 supp ∅ ) ) , ( 𝐹 supp ∅ ) ) → OrdIso ( E , ( 𝐹 supp ∅ ) ) : dom OrdIso ( E , ( 𝐹 supp ∅ ) ) –1-1-onto→ ( 𝐹 supp ∅ ) ) |
20 |
|
f1ofo |
⊢ ( OrdIso ( E , ( 𝐹 supp ∅ ) ) : dom OrdIso ( E , ( 𝐹 supp ∅ ) ) –1-1-onto→ ( 𝐹 supp ∅ ) → OrdIso ( E , ( 𝐹 supp ∅ ) ) : dom OrdIso ( E , ( 𝐹 supp ∅ ) ) –onto→ ( 𝐹 supp ∅ ) ) |
21 |
|
foima |
⊢ ( OrdIso ( E , ( 𝐹 supp ∅ ) ) : dom OrdIso ( E , ( 𝐹 supp ∅ ) ) –onto→ ( 𝐹 supp ∅ ) → ( OrdIso ( E , ( 𝐹 supp ∅ ) ) “ dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ) = ( 𝐹 supp ∅ ) ) |
22 |
18 19 20 21
|
4syl |
⊢ ( 𝜑 → ( OrdIso ( E , ( 𝐹 supp ∅ ) ) “ dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ) = ( 𝐹 supp ∅ ) ) |
23 |
22 7
|
eqsstrd |
⊢ ( 𝜑 → ( OrdIso ( E , ( 𝐹 supp ∅ ) ) “ dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ) ⊆ 𝐶 ) |
24 |
1 2 3 8 4 9 5 14 6 23
|
cantnflt |
⊢ ( 𝜑 → ( seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝐴 ↑o ( OrdIso ( E , ( 𝐹 supp ∅ ) ) ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( OrdIso ( E , ( 𝐹 supp ∅ ) ) ‘ 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) ‘ dom OrdIso ( E , ( 𝐹 supp ∅ ) ) ) ∈ ( 𝐴 ↑o 𝐶 ) ) |
25 |
10 24
|
eqeltrd |
⊢ ( 𝜑 → ( ( 𝐴 CNF 𝐵 ) ‘ 𝐹 ) ∈ ( 𝐴 ↑o 𝐶 ) ) |