Step |
Hyp |
Ref |
Expression |
1 |
|
oacomf1o.1 |
⊢ 𝐹 = ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) |
2 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) |
3 |
2
|
oacomf1olem |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) : 𝐴 –1-1-onto→ ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∧ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∩ 𝐵 ) = ∅ ) ) |
4 |
3
|
simpld |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) : 𝐴 –1-1-onto→ ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ) |
5 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) = ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) |
6 |
5
|
oacomf1olem |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵 –1-1-onto→ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∧ ( ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) = ∅ ) ) |
7 |
6
|
ancoms |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵 –1-1-onto→ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∧ ( ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) = ∅ ) ) |
8 |
7
|
simpld |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵 –1-1-onto→ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) |
9 |
|
f1ocnv |
⊢ ( ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵 –1-1-onto→ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) → ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) –1-1-onto→ 𝐵 ) |
10 |
8 9
|
syl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) –1-1-onto→ 𝐵 ) |
11 |
|
incom |
⊢ ( 𝐴 ∩ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) = ( ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) |
12 |
7
|
simprd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) = ∅ ) |
13 |
11 12
|
eqtrid |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ∩ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) = ∅ ) |
14 |
3
|
simprd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∩ 𝐵 ) = ∅ ) |
15 |
|
f1oun |
⊢ ( ( ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) : 𝐴 –1-1-onto→ ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∧ ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) –1-1-onto→ 𝐵 ) ∧ ( ( 𝐴 ∩ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) = ∅ ∧ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∩ 𝐵 ) = ∅ ) ) → ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) : ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ) |
16 |
4 10 13 14 15
|
syl22anc |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) : ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ) |
17 |
|
f1oeq1 |
⊢ ( 𝐹 = ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) → ( 𝐹 : ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ↔ ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) : ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ) ) |
18 |
1 17
|
ax-mp |
⊢ ( 𝐹 : ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ↔ ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) : ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ) |
19 |
16 18
|
sylibr |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐹 : ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ) |
20 |
|
oarec |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o 𝐵 ) = ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) ) |
21 |
20
|
f1oeq2d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐹 : ( 𝐴 +o 𝐵 ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ↔ 𝐹 : ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ) ) |
22 |
19 21
|
mpbird |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐹 : ( 𝐴 +o 𝐵 ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ) |
23 |
|
oarec |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( 𝐵 +o 𝐴 ) = ( 𝐵 ∪ ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ) ) |
24 |
23
|
ancoms |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 +o 𝐴 ) = ( 𝐵 ∪ ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ) ) |
25 |
|
uncom |
⊢ ( 𝐵 ∪ ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ) = ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) |
26 |
24 25
|
eqtrdi |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 +o 𝐴 ) = ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ) |
27 |
26
|
f1oeq3d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐹 : ( 𝐴 +o 𝐵 ) –1-1-onto→ ( 𝐵 +o 𝐴 ) ↔ 𝐹 : ( 𝐴 +o 𝐵 ) –1-1-onto→ ( ran ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 +o 𝑥 ) ) ∪ 𝐵 ) ) ) |
28 |
22 27
|
mpbird |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐹 : ( 𝐴 +o 𝐵 ) –1-1-onto→ ( 𝐵 +o 𝐴 ) ) |