Metamath Proof Explorer


Theorem fv1stcnv

Description: The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020)

Ref Expression
Assertion fv1stcnv ( ( 𝑋𝐴𝑌𝑉 ) → ( ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ‘ 𝑋 ) = ⟨ 𝑋 , 𝑌 ⟩ )

Proof

Step Hyp Ref Expression
1 snidg ( 𝑌𝑉𝑌 ∈ { 𝑌 } )
2 1 anim2i ( ( 𝑋𝐴𝑌𝑉 ) → ( 𝑋𝐴𝑌 ∈ { 𝑌 } ) )
3 eqid 𝑋 = 𝑋
4 2 3 jctir ( ( 𝑋𝐴𝑌𝑉 ) → ( ( 𝑋𝐴𝑌 ∈ { 𝑌 } ) ∧ 𝑋 = 𝑋 ) )
5 opex 𝑋 , 𝑌 ⟩ ∈ V
6 brcnvg ( ( 𝑋𝐴 ∧ ⟨ 𝑋 , 𝑌 ⟩ ∈ V ) → ( 𝑋 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ⟨ 𝑋 , 𝑌 ⟩ ↔ ⟨ 𝑋 , 𝑌 ⟩ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 𝑋 ) )
7 5 6 mpan2 ( 𝑋𝐴 → ( 𝑋 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ⟨ 𝑋 , 𝑌 ⟩ ↔ ⟨ 𝑋 , 𝑌 ⟩ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 𝑋 ) )
8 brres ( 𝑋𝐴 → ( ⟨ 𝑋 , 𝑌 ⟩ ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) 𝑋 ↔ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐴 × { 𝑌 } ) ∧ ⟨ 𝑋 , 𝑌 ⟩ 1st 𝑋 ) ) )
9 7 8 bitrd ( 𝑋𝐴 → ( 𝑋 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ⟨ 𝑋 , 𝑌 ⟩ ↔ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐴 × { 𝑌 } ) ∧ ⟨ 𝑋 , 𝑌 ⟩ 1st 𝑋 ) ) )
10 9 adantr ( ( 𝑋𝐴𝑌𝑉 ) → ( 𝑋 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ⟨ 𝑋 , 𝑌 ⟩ ↔ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐴 × { 𝑌 } ) ∧ ⟨ 𝑋 , 𝑌 ⟩ 1st 𝑋 ) ) )
11 opelxp ( ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐴 × { 𝑌 } ) ↔ ( 𝑋𝐴𝑌 ∈ { 𝑌 } ) )
12 11 anbi1i ( ( ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐴 × { 𝑌 } ) ∧ ⟨ 𝑋 , 𝑌 ⟩ 1st 𝑋 ) ↔ ( ( 𝑋𝐴𝑌 ∈ { 𝑌 } ) ∧ ⟨ 𝑋 , 𝑌 ⟩ 1st 𝑋 ) )
13 br1steqg ( ( 𝑋𝐴𝑌𝑉 ) → ( ⟨ 𝑋 , 𝑌 ⟩ 1st 𝑋𝑋 = 𝑋 ) )
14 13 anbi2d ( ( 𝑋𝐴𝑌𝑉 ) → ( ( ( 𝑋𝐴𝑌 ∈ { 𝑌 } ) ∧ ⟨ 𝑋 , 𝑌 ⟩ 1st 𝑋 ) ↔ ( ( 𝑋𝐴𝑌 ∈ { 𝑌 } ) ∧ 𝑋 = 𝑋 ) ) )
15 12 14 syl5bb ( ( 𝑋𝐴𝑌𝑉 ) → ( ( ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐴 × { 𝑌 } ) ∧ ⟨ 𝑋 , 𝑌 ⟩ 1st 𝑋 ) ↔ ( ( 𝑋𝐴𝑌 ∈ { 𝑌 } ) ∧ 𝑋 = 𝑋 ) ) )
16 10 15 bitrd ( ( 𝑋𝐴𝑌𝑉 ) → ( 𝑋 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ⟨ 𝑋 , 𝑌 ⟩ ↔ ( ( 𝑋𝐴𝑌 ∈ { 𝑌 } ) ∧ 𝑋 = 𝑋 ) ) )
17 4 16 mpbird ( ( 𝑋𝐴𝑌𝑉 ) → 𝑋 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ⟨ 𝑋 , 𝑌 ⟩ )
18 1stconst ( 𝑌𝑉 → ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) : ( 𝐴 × { 𝑌 } ) –1-1-onto𝐴 )
19 f1ocnv ( ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) : ( 𝐴 × { 𝑌 } ) –1-1-onto𝐴 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) : 𝐴1-1-onto→ ( 𝐴 × { 𝑌 } ) )
20 f1ofn ( ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) : 𝐴1-1-onto→ ( 𝐴 × { 𝑌 } ) → ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) Fn 𝐴 )
21 18 19 20 3syl ( 𝑌𝑉 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) Fn 𝐴 )
22 simpl ( ( 𝑋𝐴𝑌𝑉 ) → 𝑋𝐴 )
23 fnbrfvb ( ( ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) Fn 𝐴𝑋𝐴 ) → ( ( ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ‘ 𝑋 ) = ⟨ 𝑋 , 𝑌 ⟩ ↔ 𝑋 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ⟨ 𝑋 , 𝑌 ⟩ ) )
24 21 22 23 syl2an2 ( ( 𝑋𝐴𝑌𝑉 ) → ( ( ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ‘ 𝑋 ) = ⟨ 𝑋 , 𝑌 ⟩ ↔ 𝑋 ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ⟨ 𝑋 , 𝑌 ⟩ ) )
25 17 24 mpbird ( ( 𝑋𝐴𝑌𝑉 ) → ( ( 1st ↾ ( 𝐴 × { 𝑌 } ) ) ‘ 𝑋 ) = ⟨ 𝑋 , 𝑌 ⟩ )