Step |
Hyp |
Ref |
Expression |
1 |
|
noreson |
⊢ ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ) → ( 𝐴 ↾ 𝑋 ) ∈ No ) |
2 |
1
|
3adant2 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( 𝐴 ↾ 𝑋 ) ∈ No ) |
3 |
|
noreson |
⊢ ( ( 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( 𝐵 ↾ 𝑋 ) ∈ No ) |
4 |
3
|
3adant1 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( 𝐵 ↾ 𝑋 ) ∈ No ) |
5 |
|
sltintdifex |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ∈ No ∧ ( 𝐵 ↾ 𝑋 ) ∈ No ) → ( ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ V ) ) |
6 |
|
onintrab |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ V ↔ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ On ) |
7 |
5 6
|
syl6ib |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ∈ No ∧ ( 𝐵 ↾ 𝑋 ) ∈ No ) → ( ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ On ) ) |
8 |
2 4 7
|
syl2anc |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ On ) ) |
9 |
8
|
imp |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ On ) |
10 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) → 𝑋 ∈ On ) |
11 |
|
sltval2 |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ∈ No ∧ ( 𝐵 ↾ 𝑋 ) ∈ No ) → ( ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ↔ ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) ) |
12 |
2 4 11
|
syl2anc |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ↔ ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) ) |
13 |
|
fvex |
⊢ ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ V |
14 |
|
fvex |
⊢ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ V |
15 |
13 14
|
brtp |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ↔ ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ∨ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) ) |
16 |
|
1n0 |
⊢ 1o ≠ ∅ |
17 |
16
|
neii |
⊢ ¬ 1o = ∅ |
18 |
|
eqeq1 |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o → ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ↔ 1o = ∅ ) ) |
19 |
17 18
|
mtbiri |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o → ¬ ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
20 |
|
ndmfv |
⊢ ( ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
21 |
19 20
|
nsyl2 |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ) |
22 |
21
|
adantr |
⊢ ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ) |
23 |
22
|
orcd |
⊢ ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ∨ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) ) |
24 |
21
|
adantr |
⊢ ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ) |
25 |
24
|
orcd |
⊢ ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ∨ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) ) |
26 |
|
2on |
⊢ 2o ∈ On |
27 |
26
|
elexi |
⊢ 2o ∈ V |
28 |
27
|
prid2 |
⊢ 2o ∈ { 1o , 2o } |
29 |
28
|
nosgnn0i |
⊢ ∅ ≠ 2o |
30 |
29
|
neii |
⊢ ¬ ∅ = 2o |
31 |
|
eqeq1 |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o → ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ↔ 2o = ∅ ) ) |
32 |
|
eqcom |
⊢ ( 2o = ∅ ↔ ∅ = 2o ) |
33 |
31 32
|
bitrdi |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o → ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ↔ ∅ = 2o ) ) |
34 |
30 33
|
mtbiri |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o → ¬ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
35 |
|
ndmfv |
⊢ ( ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) → ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
36 |
34 35
|
nsyl2 |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) |
37 |
36
|
adantl |
⊢ ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) |
38 |
37
|
olcd |
⊢ ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ∨ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) ) |
39 |
23 25 38
|
3jaoi |
⊢ ( ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ∨ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ∨ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) ) |
40 |
15 39
|
sylbi |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ∨ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) ) |
41 |
12 40
|
syl6bi |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ∨ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) ) ) |
42 |
41
|
imp |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ∨ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) ) |
43 |
|
dmres |
⊢ dom ( 𝐴 ↾ 𝑋 ) = ( 𝑋 ∩ dom 𝐴 ) |
44 |
43
|
elin2 |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ↔ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 ∧ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐴 ) ) |
45 |
44
|
simplbi |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 ) |
46 |
|
dmres |
⊢ dom ( 𝐵 ↾ 𝑋 ) = ( 𝑋 ∩ dom 𝐵 ) |
47 |
46
|
elin2 |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ↔ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 ∧ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐵 ) ) |
48 |
47
|
simplbi |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 ) |
49 |
45 48
|
jaoi |
⊢ ( ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ∨ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 ) |
50 |
42 49
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 ) |
51 |
|
onelss |
⊢ ( 𝑋 ∈ On → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ⊆ 𝑋 ) ) |
52 |
10 50 51
|
sylc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ⊆ 𝑋 ) |
53 |
52
|
sselda |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → 𝑦 ∈ 𝑋 ) |
54 |
|
onelon |
⊢ ( ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ On ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → 𝑦 ∈ On ) |
55 |
9 54
|
sylan |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → 𝑦 ∈ On ) |
56 |
|
intss1 |
⊢ ( 𝑦 ∈ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ⊆ 𝑦 ) |
57 |
|
ontri1 |
⊢ ( ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ On ∧ 𝑦 ∈ On ) → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ⊆ 𝑦 ↔ ¬ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
58 |
56 57
|
syl5ib |
⊢ ( ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ On ∧ 𝑦 ∈ On ) → ( 𝑦 ∈ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } → ¬ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
59 |
58
|
con2d |
⊢ ( ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ On ∧ 𝑦 ∈ On ) → ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } → ¬ 𝑦 ∈ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
60 |
9 59
|
sylan |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) ∧ 𝑦 ∈ On ) → ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } → ¬ 𝑦 ∈ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
61 |
60
|
impancom |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → ( 𝑦 ∈ On → ¬ 𝑦 ∈ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
62 |
55 61
|
mpd |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → ¬ 𝑦 ∈ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) |
63 |
|
fveq2 |
⊢ ( 𝑎 = 𝑦 → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) = ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) ) |
64 |
|
fveq2 |
⊢ ( 𝑎 = 𝑦 → ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) |
65 |
63 64
|
neeq12d |
⊢ ( 𝑎 = 𝑦 → ( ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) ↔ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) ) |
66 |
65
|
elrab |
⊢ ( 𝑦 ∈ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ↔ ( 𝑦 ∈ On ∧ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) ) |
67 |
66
|
simplbi2 |
⊢ ( 𝑦 ∈ On → ( ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) → 𝑦 ∈ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
68 |
67
|
con3d |
⊢ ( 𝑦 ∈ On → ( ¬ 𝑦 ∈ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } → ¬ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) ) |
69 |
55 62 68
|
sylc |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → ¬ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) |
70 |
|
df-ne |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ↔ ¬ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) |
71 |
70
|
con2bii |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ↔ ¬ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) |
72 |
69 71
|
sylibr |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) |
73 |
|
fvres |
⊢ ( 𝑦 ∈ 𝑋 → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ) |
74 |
|
fvres |
⊢ ( 𝑦 ∈ 𝑋 → ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
75 |
73 74
|
eqeq12d |
⊢ ( 𝑦 ∈ 𝑋 → ( ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ↔ ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ) |
76 |
75
|
biimpd |
⊢ ( 𝑦 ∈ 𝑋 → ( ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) → ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ) |
77 |
53 72 76
|
sylc |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
78 |
77
|
ralrimiva |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) → ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
79 |
|
fvresval |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∨ ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
80 |
79
|
ori |
⊢ ( ¬ ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
81 |
19 80
|
nsyl2 |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o → ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
82 |
81
|
eqcomd |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
83 |
|
eqeq2 |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ) ) |
84 |
82 83
|
mpbid |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ) |
85 |
84
|
adantr |
⊢ ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ) |
86 |
85
|
a1i |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ) ) |
87 |
21
|
ad2antrl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ) |
88 |
87 45
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 ) |
89 |
|
nofun |
⊢ ( ( 𝐵 ↾ 𝑋 ) ∈ No → Fun ( 𝐵 ↾ 𝑋 ) ) |
90 |
|
fvelrn |
⊢ ( ( Fun ( 𝐵 ↾ 𝑋 ) ∧ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) → ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ ran ( 𝐵 ↾ 𝑋 ) ) |
91 |
90
|
ex |
⊢ ( Fun ( 𝐵 ↾ 𝑋 ) → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) → ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ ran ( 𝐵 ↾ 𝑋 ) ) ) |
92 |
89 91
|
syl |
⊢ ( ( 𝐵 ↾ 𝑋 ) ∈ No → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) → ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ ran ( 𝐵 ↾ 𝑋 ) ) ) |
93 |
|
norn |
⊢ ( ( 𝐵 ↾ 𝑋 ) ∈ No → ran ( 𝐵 ↾ 𝑋 ) ⊆ { 1o , 2o } ) |
94 |
93
|
sseld |
⊢ ( ( 𝐵 ↾ 𝑋 ) ∈ No → ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ ran ( 𝐵 ↾ 𝑋 ) → ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ { 1o , 2o } ) ) |
95 |
92 94
|
syld |
⊢ ( ( 𝐵 ↾ 𝑋 ) ∈ No → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) → ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ { 1o , 2o } ) ) |
96 |
|
nosgnn0 |
⊢ ¬ ∅ ∈ { 1o , 2o } |
97 |
|
eleq1 |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ → ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ { 1o , 2o } ↔ ∅ ∈ { 1o , 2o } ) ) |
98 |
96 97
|
mtbiri |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ → ¬ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ { 1o , 2o } ) |
99 |
95 98
|
nsyli |
⊢ ( ( 𝐵 ↾ 𝑋 ) ∈ No → ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) ) |
100 |
4 99
|
syl |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) ) |
101 |
100
|
imp |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) |
102 |
101
|
adantrl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ) → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) |
103 |
47
|
simplbi2 |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐵 → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) ) |
104 |
103
|
con3d |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 → ( ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐵 ) ) |
105 |
88 102 104
|
sylc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ) → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐵 ) |
106 |
|
ndmfv |
⊢ ( ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐵 → ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
107 |
105 106
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ) → ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
108 |
107
|
ex |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) → ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ) |
109 |
86 108
|
jcad |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ) ) |
110 |
|
fvresval |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∨ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
111 |
110
|
ori |
⊢ ( ¬ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
112 |
34 111
|
nsyl2 |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o → ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
113 |
112
|
eqcomd |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o → ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
114 |
|
eqeq2 |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o → ( ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ↔ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) |
115 |
113 114
|
mpbid |
⊢ ( ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o → ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) |
116 |
84 115
|
anim12i |
⊢ ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) |
117 |
116
|
a1i |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) ) |
118 |
36
|
ad2antll |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐵 ↾ 𝑋 ) ) |
119 |
118 48
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 ) |
120 |
|
nofun |
⊢ ( ( 𝐴 ↾ 𝑋 ) ∈ No → Fun ( 𝐴 ↾ 𝑋 ) ) |
121 |
|
fvelrn |
⊢ ( ( Fun ( 𝐴 ↾ 𝑋 ) ∧ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ) → ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ ran ( 𝐴 ↾ 𝑋 ) ) |
122 |
121
|
ex |
⊢ ( Fun ( 𝐴 ↾ 𝑋 ) → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ ran ( 𝐴 ↾ 𝑋 ) ) ) |
123 |
120 122
|
syl |
⊢ ( ( 𝐴 ↾ 𝑋 ) ∈ No → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ ran ( 𝐴 ↾ 𝑋 ) ) ) |
124 |
|
norn |
⊢ ( ( 𝐴 ↾ 𝑋 ) ∈ No → ran ( 𝐴 ↾ 𝑋 ) ⊆ { 1o , 2o } ) |
125 |
124
|
sseld |
⊢ ( ( 𝐴 ↾ 𝑋 ) ∈ No → ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ ran ( 𝐴 ↾ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ { 1o , 2o } ) ) |
126 |
123 125
|
syld |
⊢ ( ( 𝐴 ↾ 𝑋 ) ∈ No → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ { 1o , 2o } ) ) |
127 |
|
eleq1 |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ → ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ { 1o , 2o } ↔ ∅ ∈ { 1o , 2o } ) ) |
128 |
96 127
|
mtbiri |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ → ¬ ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ { 1o , 2o } ) |
129 |
126 128
|
nsyli |
⊢ ( ( 𝐴 ↾ 𝑋 ) ∈ No → ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ) ) |
130 |
2 129
|
syl |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ) ) |
131 |
130
|
imp |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ) |
132 |
131
|
adantrr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ) |
133 |
44
|
simplbi2 |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 → ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐴 → ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) ) ) |
134 |
133
|
con3d |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ 𝑋 → ( ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom ( 𝐴 ↾ 𝑋 ) → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐴 ) ) |
135 |
119 132 134
|
sylc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐴 ) |
136 |
135
|
ex |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐴 ) ) |
137 |
|
ndmfv |
⊢ ( ¬ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ dom 𝐴 → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) |
138 |
136 137
|
syl6 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ) |
139 |
115
|
adantl |
⊢ ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) |
140 |
139
|
a1i |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) |
141 |
138 140
|
jcad |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) ) |
142 |
109 117 141
|
3orim123d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ∨ ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) → ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) ) ) |
143 |
|
fvex |
⊢ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ V |
144 |
|
fvex |
⊢ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ∈ V |
145 |
143 144
|
brtp |
⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ↔ ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) = 2o ) ) ) |
146 |
142 15 145
|
3imtr4g |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( ( 𝐴 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐵 ↾ 𝑋 ) ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) ) |
147 |
12 146
|
sylbid |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) ) |
148 |
147
|
imp |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
149 |
|
raleq |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ) |
150 |
|
fveq2 |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } → ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
151 |
|
fveq2 |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) |
152 |
150 151
|
breq12d |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } → ( ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) ) |
153 |
149 152
|
anbi12d |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) ) ) |
154 |
153
|
rspcev |
⊢ ( ( ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ∈ On ∧ ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑎 ) ≠ ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑎 ) } ) ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) |
155 |
9 78 148 154
|
syl12anc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) |
156 |
|
sltval |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) ) |
157 |
156
|
3adant3 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( 𝐴 <s 𝐵 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) ) |
158 |
157
|
adantr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) → ( 𝐴 <s 𝐵 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) ) |
159 |
155 158
|
mpbird |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) ) → 𝐴 <s 𝐵 ) |
160 |
159
|
ex |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) → ( ( 𝐴 ↾ 𝑋 ) <s ( 𝐵 ↾ 𝑋 ) → 𝐴 <s 𝐵 ) ) |