Step |
Hyp |
Ref |
Expression |
1 |
|
omeulem1 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) |
2 |
|
opex |
⊢ 〈 𝑥 , 𝑦 〉 ∈ V |
3 |
2
|
isseti |
⊢ ∃ 𝑧 𝑧 = 〈 𝑥 , 𝑦 〉 |
4 |
|
19.41v |
⊢ ( ∃ 𝑧 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ↔ ( ∃ 𝑧 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) |
5 |
3 4
|
mpbiran |
⊢ ( ∃ 𝑧 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ↔ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) |
6 |
5
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐴 ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) |
7 |
|
rexcom4 |
⊢ ( ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ↔ ∃ 𝑧 ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) |
8 |
6 7
|
bitr3i |
⊢ ( ∃ 𝑦 ∈ 𝐴 ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ↔ ∃ 𝑧 ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) |
9 |
8
|
rexbii |
⊢ ( ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ↔ ∃ 𝑥 ∈ On ∃ 𝑧 ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) |
10 |
|
rexcom4 |
⊢ ( ∃ 𝑥 ∈ On ∃ 𝑧 ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ↔ ∃ 𝑧 ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) |
11 |
9 10
|
bitri |
⊢ ( ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ↔ ∃ 𝑧 ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) |
12 |
1 11
|
sylib |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → ∃ 𝑧 ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) |
13 |
|
simp2rl |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 𝑧 = 〈 𝑥 , 𝑦 〉 ) |
14 |
|
simp3rl |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 𝑡 = 〈 𝑟 , 𝑠 〉 ) |
15 |
|
simp2rr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) |
16 |
|
simp3rr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) |
17 |
15 16
|
eqtr4d |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) ) |
18 |
|
simp11 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 𝐴 ∈ On ) |
19 |
|
simp13 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 𝐴 ≠ ∅ ) |
20 |
|
simp2ll |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 𝑥 ∈ On ) |
21 |
|
simp2lr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 𝑦 ∈ 𝐴 ) |
22 |
|
simp3ll |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 𝑟 ∈ On ) |
23 |
|
simp3lr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 𝑠 ∈ 𝐴 ) |
24 |
|
omopth2 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ) → ( ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) ↔ ( 𝑥 = 𝑟 ∧ 𝑦 = 𝑠 ) ) ) |
25 |
18 19 20 21 22 23 24
|
syl222anc |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → ( ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) ↔ ( 𝑥 = 𝑟 ∧ 𝑦 = 𝑠 ) ) ) |
26 |
17 25
|
mpbid |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → ( 𝑥 = 𝑟 ∧ 𝑦 = 𝑠 ) ) |
27 |
|
opeq12 |
⊢ ( ( 𝑥 = 𝑟 ∧ 𝑦 = 𝑠 ) → 〈 𝑥 , 𝑦 〉 = 〈 𝑟 , 𝑠 〉 ) |
28 |
26 27
|
syl |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 〈 𝑥 , 𝑦 〉 = 〈 𝑟 , 𝑠 〉 ) |
29 |
14 28
|
eqtr4d |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 𝑡 = 〈 𝑥 , 𝑦 〉 ) |
30 |
13 29
|
eqtr4d |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ∧ ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) → 𝑧 = 𝑡 ) |
31 |
30
|
3expia |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) ) → ( ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) ∧ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) → 𝑧 = 𝑡 ) ) |
32 |
31
|
exp4b |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → ( ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) → ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) → ( ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) → 𝑧 = 𝑡 ) ) ) ) |
33 |
32
|
expd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) → ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) → ( ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) → 𝑧 = 𝑡 ) ) ) ) ) |
34 |
33
|
rexlimdvv |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → ( ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) → ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) → ( ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) → 𝑧 = 𝑡 ) ) ) ) |
35 |
34
|
imp |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) → ( ( 𝑟 ∈ On ∧ 𝑠 ∈ 𝐴 ) → ( ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) → 𝑧 = 𝑡 ) ) ) |
36 |
35
|
rexlimdvv |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) ∧ ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) → ( ∃ 𝑟 ∈ On ∃ 𝑠 ∈ 𝐴 ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) → 𝑧 = 𝑡 ) ) |
37 |
36
|
expimpd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → ( ( ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ∧ ∃ 𝑟 ∈ On ∃ 𝑠 ∈ 𝐴 ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) → 𝑧 = 𝑡 ) ) |
38 |
37
|
alrimivv |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → ∀ 𝑧 ∀ 𝑡 ( ( ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ∧ ∃ 𝑟 ∈ On ∃ 𝑠 ∈ 𝐴 ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) → 𝑧 = 𝑡 ) ) |
39 |
|
opeq1 |
⊢ ( 𝑥 = 𝑟 → 〈 𝑥 , 𝑦 〉 = 〈 𝑟 , 𝑦 〉 ) |
40 |
39
|
eqeq2d |
⊢ ( 𝑥 = 𝑟 → ( 𝑧 = 〈 𝑥 , 𝑦 〉 ↔ 𝑧 = 〈 𝑟 , 𝑦 〉 ) ) |
41 |
|
oveq2 |
⊢ ( 𝑥 = 𝑟 → ( 𝐴 ·o 𝑥 ) = ( 𝐴 ·o 𝑟 ) ) |
42 |
41
|
oveq1d |
⊢ ( 𝑥 = 𝑟 → ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = ( ( 𝐴 ·o 𝑟 ) +o 𝑦 ) ) |
43 |
42
|
eqeq1d |
⊢ ( 𝑥 = 𝑟 → ( ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ↔ ( ( 𝐴 ·o 𝑟 ) +o 𝑦 ) = 𝐵 ) ) |
44 |
40 43
|
anbi12d |
⊢ ( 𝑥 = 𝑟 → ( ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ↔ ( 𝑧 = 〈 𝑟 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑦 ) = 𝐵 ) ) ) |
45 |
|
opeq2 |
⊢ ( 𝑦 = 𝑠 → 〈 𝑟 , 𝑦 〉 = 〈 𝑟 , 𝑠 〉 ) |
46 |
45
|
eqeq2d |
⊢ ( 𝑦 = 𝑠 → ( 𝑧 = 〈 𝑟 , 𝑦 〉 ↔ 𝑧 = 〈 𝑟 , 𝑠 〉 ) ) |
47 |
|
oveq2 |
⊢ ( 𝑦 = 𝑠 → ( ( 𝐴 ·o 𝑟 ) +o 𝑦 ) = ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) ) |
48 |
47
|
eqeq1d |
⊢ ( 𝑦 = 𝑠 → ( ( ( 𝐴 ·o 𝑟 ) +o 𝑦 ) = 𝐵 ↔ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) |
49 |
46 48
|
anbi12d |
⊢ ( 𝑦 = 𝑠 → ( ( 𝑧 = 〈 𝑟 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑦 ) = 𝐵 ) ↔ ( 𝑧 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) |
50 |
44 49
|
cbvrex2vw |
⊢ ( ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ↔ ∃ 𝑟 ∈ On ∃ 𝑠 ∈ 𝐴 ( 𝑧 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) |
51 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑡 → ( 𝑧 = 〈 𝑟 , 𝑠 〉 ↔ 𝑡 = 〈 𝑟 , 𝑠 〉 ) ) |
52 |
51
|
anbi1d |
⊢ ( 𝑧 = 𝑡 → ( ( 𝑧 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ↔ ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) |
53 |
52
|
2rexbidv |
⊢ ( 𝑧 = 𝑡 → ( ∃ 𝑟 ∈ On ∃ 𝑠 ∈ 𝐴 ( 𝑧 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ↔ ∃ 𝑟 ∈ On ∃ 𝑠 ∈ 𝐴 ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) |
54 |
50 53
|
bitrid |
⊢ ( 𝑧 = 𝑡 → ( ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ↔ ∃ 𝑟 ∈ On ∃ 𝑠 ∈ 𝐴 ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) ) |
55 |
54
|
eu4 |
⊢ ( ∃! 𝑧 ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ↔ ( ∃ 𝑧 ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ∧ ∀ 𝑧 ∀ 𝑡 ( ( ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ∧ ∃ 𝑟 ∈ On ∃ 𝑠 ∈ 𝐴 ( 𝑡 = 〈 𝑟 , 𝑠 〉 ∧ ( ( 𝐴 ·o 𝑟 ) +o 𝑠 ) = 𝐵 ) ) → 𝑧 = 𝑡 ) ) ) |
56 |
12 38 55
|
sylanbrc |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → ∃! 𝑧 ∃ 𝑥 ∈ On ∃ 𝑦 ∈ 𝐴 ( 𝑧 = 〈 𝑥 , 𝑦 〉 ∧ ( ( 𝐴 ·o 𝑥 ) +o 𝑦 ) = 𝐵 ) ) |