Step |
Hyp |
Ref |
Expression |
1 |
|
oicl.1 |
⊢ 𝐹 = OrdIso ( 𝑅 , 𝐴 ) |
2 |
|
eqid |
⊢ recs ( ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ∀ 𝑢 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ¬ 𝑢 𝑅 𝑣 ) ) ) = recs ( ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ∀ 𝑢 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ¬ 𝑢 𝑅 𝑣 ) ) ) |
3 |
|
eqid |
⊢ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } = { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } |
4 |
|
eqid |
⊢ ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ∀ 𝑢 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ¬ 𝑢 𝑅 𝑣 ) ) = ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ∀ 𝑢 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ¬ 𝑢 𝑅 𝑣 ) ) |
5 |
2 3 4
|
ordtypecbv |
⊢ recs ( ( 𝑓 ∈ V ↦ ( ℩ 𝑠 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ∀ 𝑟 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ¬ 𝑟 𝑅 𝑠 ) ) ) = recs ( ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ∀ 𝑢 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ¬ 𝑢 𝑅 𝑣 ) ) ) |
6 |
|
eqid |
⊢ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( recs ( ( 𝑓 ∈ V ↦ ( ℩ 𝑠 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ∀ 𝑟 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ¬ 𝑟 𝑅 𝑠 ) ) ) “ 𝑥 ) 𝑧 𝑅 𝑡 } = { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( recs ( ( 𝑓 ∈ V ↦ ( ℩ 𝑠 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ∀ 𝑟 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ¬ 𝑟 𝑅 𝑠 ) ) ) “ 𝑥 ) 𝑧 𝑅 𝑡 } |
7 |
|
simpl |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 We 𝐴 ) |
8 |
|
simpr |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Se 𝐴 ) |
9 |
5 3 4 6 1 7 8
|
ordtypelem5 |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( Ord dom 𝐹 ∧ 𝐹 : dom 𝐹 ⟶ 𝐴 ) ) |
10 |
9
|
simpld |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → Ord dom 𝐹 ) |
11 |
|
ord0 |
⊢ Ord ∅ |
12 |
1
|
oi0 |
⊢ ( ¬ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 = ∅ ) |
13 |
12
|
dmeqd |
⊢ ( ¬ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → dom 𝐹 = dom ∅ ) |
14 |
|
dm0 |
⊢ dom ∅ = ∅ |
15 |
13 14
|
eqtrdi |
⊢ ( ¬ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → dom 𝐹 = ∅ ) |
16 |
|
ordeq |
⊢ ( dom 𝐹 = ∅ → ( Ord dom 𝐹 ↔ Ord ∅ ) ) |
17 |
15 16
|
syl |
⊢ ( ¬ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( Ord dom 𝐹 ↔ Ord ∅ ) ) |
18 |
11 17
|
mpbiri |
⊢ ( ¬ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → Ord dom 𝐹 ) |
19 |
10 18
|
pm2.61i |
⊢ Ord dom 𝐹 |