Step |
Hyp |
Ref |
Expression |
1 |
|
xp1st |
⊢ ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) → ( 1st ‘ 𝐴 ) ∈ ( 𝑈 × 𝑉 ) ) |
2 |
1
|
ad2antrl |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) → ( 1st ‘ 𝐴 ) ∈ ( 𝑈 × 𝑉 ) ) |
3 |
|
3simpa |
⊢ ( ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) → ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) → ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ) ) |
5 |
4
|
adantl |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) → ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ) ) |
6 |
|
eqopi |
⊢ ( ( ( 1st ‘ 𝐴 ) ∈ ( 𝑈 × 𝑉 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ) ) → ( 1st ‘ 𝐴 ) = 〈 𝑋 , 𝑌 〉 ) |
7 |
2 5 6
|
syl2anc |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) → ( 1st ‘ 𝐴 ) = 〈 𝑋 , 𝑌 〉 ) |
8 |
|
simprr3 |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) → ( 2nd ‘ 𝐴 ) = 𝑍 ) |
9 |
7 8
|
jca |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) → ( ( 1st ‘ 𝐴 ) = 〈 𝑋 , 𝑌 〉 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) |
10 |
|
df-ot |
⊢ 〈 𝑋 , 𝑌 , 𝑍 〉 = 〈 〈 𝑋 , 𝑌 〉 , 𝑍 〉 |
11 |
10
|
eqeq2i |
⊢ ( 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ↔ 𝐴 = 〈 〈 𝑋 , 𝑌 〉 , 𝑍 〉 ) |
12 |
|
eqop |
⊢ ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) → ( 𝐴 = 〈 〈 𝑋 , 𝑌 〉 , 𝑍 〉 ↔ ( ( 1st ‘ 𝐴 ) = 〈 𝑋 , 𝑌 〉 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) |
13 |
11 12
|
syl5bb |
⊢ ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) → ( 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ↔ ( ( 1st ‘ 𝐴 ) = 〈 𝑋 , 𝑌 〉 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) |
14 |
13
|
ad2antrl |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) → ( 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ↔ ( ( 1st ‘ 𝐴 ) = 〈 𝑋 , 𝑌 〉 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) |
15 |
9 14
|
mpbird |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) → 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ) |
16 |
|
opelxpi |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ) → 〈 𝑋 , 𝑌 〉 ∈ ( 𝑈 × 𝑉 ) ) |
17 |
16
|
3adant3 |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → 〈 𝑋 , 𝑌 〉 ∈ ( 𝑈 × 𝑉 ) ) |
18 |
|
simp3 |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → 𝑍 ∈ 𝑊 ) |
19 |
17 18
|
opelxpd |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → 〈 〈 𝑋 , 𝑌 〉 , 𝑍 〉 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ) |
20 |
10 19
|
eqeltrid |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → 〈 𝑋 , 𝑌 , 𝑍 〉 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ) |
21 |
20
|
adantr |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ) → 〈 𝑋 , 𝑌 , 𝑍 〉 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ) |
22 |
|
eleq1 |
⊢ ( 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 → ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ↔ 〈 𝑋 , 𝑌 , 𝑍 〉 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ) ) |
23 |
22
|
adantl |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ) → ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ↔ 〈 𝑋 , 𝑌 , 𝑍 〉 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ) ) |
24 |
21 23
|
mpbird |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ) → 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ) |
25 |
|
2fveq3 |
⊢ ( 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 → ( 1st ‘ ( 1st ‘ 𝐴 ) ) = ( 1st ‘ ( 1st ‘ 〈 𝑋 , 𝑌 , 𝑍 〉 ) ) ) |
26 |
|
ot1stg |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 1st ‘ ( 1st ‘ 〈 𝑋 , 𝑌 , 𝑍 〉 ) ) = 𝑋 ) |
27 |
25 26
|
sylan9eqr |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ) → ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ) |
28 |
|
2fveq3 |
⊢ ( 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 → ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = ( 2nd ‘ ( 1st ‘ 〈 𝑋 , 𝑌 , 𝑍 〉 ) ) ) |
29 |
|
ot2ndg |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 2nd ‘ ( 1st ‘ 〈 𝑋 , 𝑌 , 𝑍 〉 ) ) = 𝑌 ) |
30 |
28 29
|
sylan9eqr |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ) → ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ) |
31 |
|
fveq2 |
⊢ ( 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 → ( 2nd ‘ 𝐴 ) = ( 2nd ‘ 〈 𝑋 , 𝑌 , 𝑍 〉 ) ) |
32 |
|
ot3rdg |
⊢ ( 𝑍 ∈ 𝑊 → ( 2nd ‘ 〈 𝑋 , 𝑌 , 𝑍 〉 ) = 𝑍 ) |
33 |
32
|
3ad2ant3 |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 2nd ‘ 〈 𝑋 , 𝑌 , 𝑍 〉 ) = 𝑍 ) |
34 |
31 33
|
sylan9eqr |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ) → ( 2nd ‘ 𝐴 ) = 𝑍 ) |
35 |
27 30 34
|
3jca |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ) → ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) |
36 |
24 35
|
jca |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ) → ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ) |
37 |
15 36
|
impbida |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( ( 𝐴 ∈ ( ( 𝑈 × 𝑉 ) × 𝑊 ) ∧ ( ( 1st ‘ ( 1st ‘ 𝐴 ) ) = 𝑋 ∧ ( 2nd ‘ ( 1st ‘ 𝐴 ) ) = 𝑌 ∧ ( 2nd ‘ 𝐴 ) = 𝑍 ) ) ↔ 𝐴 = 〈 𝑋 , 𝑌 , 𝑍 〉 ) ) |