Step |
Hyp |
Ref |
Expression |
1 |
|
snidg |
⊢ ( 𝑌 ∈ 𝑉 → 𝑌 ∈ { 𝑌 } ) |
2 |
1
|
anim2i |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ { 𝑌 } ) ) |
3 |
|
eqid |
⊢ 𝑋 = 𝑋 |
4 |
2 3
|
jctir |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ { 𝑌 } ) ∧ 𝑋 = 𝑋 ) ) |
5 |
|
opex |
⊢ 〈 𝑋 , 𝑌 〉 ∈ V |
6 |
|
brcnvg |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 〈 𝑋 , 𝑌 〉 ∈ V ) → ( 𝑋 ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 〈 𝑋 , 𝑌 〉 ↔ 〈 𝑋 , 𝑌 〉 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 𝑋 ) ) |
7 |
5 6
|
mpan2 |
⊢ ( 𝑋 ∈ 𝐴 → ( 𝑋 ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 〈 𝑋 , 𝑌 〉 ↔ 〈 𝑋 , 𝑌 〉 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 𝑋 ) ) |
8 |
|
brres |
⊢ ( 𝑋 ∈ 𝐴 → ( 〈 𝑋 , 𝑌 〉 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 𝑋 ↔ ( 〈 𝑋 , 𝑌 〉 ∈ ( 𝐴 × { 𝑌 } ) ∧ 〈 𝑋 , 𝑌 〉 1st 𝑋 ) ) ) |
9 |
7 8
|
bitrd |
⊢ ( 𝑋 ∈ 𝐴 → ( 𝑋 ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 〈 𝑋 , 𝑌 〉 ↔ ( 〈 𝑋 , 𝑌 〉 ∈ ( 𝐴 × { 𝑌 } ) ∧ 〈 𝑋 , 𝑌 〉 1st 𝑋 ) ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 〈 𝑋 , 𝑌 〉 ↔ ( 〈 𝑋 , 𝑌 〉 ∈ ( 𝐴 × { 𝑌 } ) ∧ 〈 𝑋 , 𝑌 〉 1st 𝑋 ) ) ) |
11 |
|
opelxp |
⊢ ( 〈 𝑋 , 𝑌 〉 ∈ ( 𝐴 × { 𝑌 } ) ↔ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ { 𝑌 } ) ) |
12 |
11
|
anbi1i |
⊢ ( ( 〈 𝑋 , 𝑌 〉 ∈ ( 𝐴 × { 𝑌 } ) ∧ 〈 𝑋 , 𝑌 〉 1st 𝑋 ) ↔ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ { 𝑌 } ) ∧ 〈 𝑋 , 𝑌 〉 1st 𝑋 ) ) |
13 |
|
br1steqg |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → ( 〈 𝑋 , 𝑌 〉 1st 𝑋 ↔ 𝑋 = 𝑋 ) ) |
14 |
13
|
anbi2d |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ { 𝑌 } ) ∧ 〈 𝑋 , 𝑌 〉 1st 𝑋 ) ↔ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ { 𝑌 } ) ∧ 𝑋 = 𝑋 ) ) ) |
15 |
12 14
|
syl5bb |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → ( ( 〈 𝑋 , 𝑌 〉 ∈ ( 𝐴 × { 𝑌 } ) ∧ 〈 𝑋 , 𝑌 〉 1st 𝑋 ) ↔ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ { 𝑌 } ) ∧ 𝑋 = 𝑋 ) ) ) |
16 |
10 15
|
bitrd |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 〈 𝑋 , 𝑌 〉 ↔ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ { 𝑌 } ) ∧ 𝑋 = 𝑋 ) ) ) |
17 |
4 16
|
mpbird |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → 𝑋 ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 〈 𝑋 , 𝑌 〉 ) |
18 |
|
1stconst |
⊢ ( 𝑌 ∈ 𝑉 → ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) : ( 𝐴 × { 𝑌 } ) –1-1-onto→ 𝐴 ) |
19 |
|
f1ocnv |
⊢ ( ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) : ( 𝐴 × { 𝑌 } ) –1-1-onto→ 𝐴 → ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) : 𝐴 –1-1-onto→ ( 𝐴 × { 𝑌 } ) ) |
20 |
|
f1ofn |
⊢ ( ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) : 𝐴 –1-1-onto→ ( 𝐴 × { 𝑌 } ) → ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) Fn 𝐴 ) |
21 |
18 19 20
|
3syl |
⊢ ( 𝑌 ∈ 𝑉 → ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) Fn 𝐴 ) |
22 |
|
simpl |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → 𝑋 ∈ 𝐴 ) |
23 |
|
fnbrfvb |
⊢ ( ( ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ( ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ‘ 𝑋 ) = 〈 𝑋 , 𝑌 〉 ↔ 𝑋 ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 〈 𝑋 , 𝑌 〉 ) ) |
24 |
21 22 23
|
syl2an2 |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → ( ( ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ‘ 𝑋 ) = 〈 𝑋 , 𝑌 〉 ↔ 𝑋 ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 〈 𝑋 , 𝑌 〉 ) ) |
25 |
17 24
|
mpbird |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉 ) → ( ◡ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ‘ 𝑋 ) = 〈 𝑋 , 𝑌 〉 ) |