Metamath Proof Explorer


Theorem mexval2

Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a list of constants and variables. (Contributed by Mario Carneiro, 18-Jul-2016)

Ref Expression
Hypotheses mexval.k 𝐾 = ( mTC ‘ 𝑇 )
mexval.e 𝐸 = ( mEx ‘ 𝑇 )
mexval2.c 𝐶 = ( mCN ‘ 𝑇 )
mexval2.v 𝑉 = ( mVR ‘ 𝑇 )
Assertion mexval2 𝐸 = ( 𝐾 × Word ( 𝐶𝑉 ) )

Proof

Step Hyp Ref Expression
1 mexval.k 𝐾 = ( mTC ‘ 𝑇 )
2 mexval.e 𝐸 = ( mEx ‘ 𝑇 )
3 mexval2.c 𝐶 = ( mCN ‘ 𝑇 )
4 mexval2.v 𝑉 = ( mVR ‘ 𝑇 )
5 eqid ( mREx ‘ 𝑇 ) = ( mREx ‘ 𝑇 )
6 1 2 5 mexval 𝐸 = ( 𝐾 × ( mREx ‘ 𝑇 ) )
7 3 4 5 mrexval ( 𝑇 ∈ V → ( mREx ‘ 𝑇 ) = Word ( 𝐶𝑉 ) )
8 7 xpeq2d ( 𝑇 ∈ V → ( 𝐾 × ( mREx ‘ 𝑇 ) ) = ( 𝐾 × Word ( 𝐶𝑉 ) ) )
9 6 8 syl5eq ( 𝑇 ∈ V → 𝐸 = ( 𝐾 × Word ( 𝐶𝑉 ) ) )
10 0xp ( ∅ × Word ( 𝐶𝑉 ) ) = ∅
11 10 eqcomi ∅ = ( ∅ × Word ( 𝐶𝑉 ) )
12 fvprc ( ¬ 𝑇 ∈ V → ( mEx ‘ 𝑇 ) = ∅ )
13 2 12 syl5eq ( ¬ 𝑇 ∈ V → 𝐸 = ∅ )
14 fvprc ( ¬ 𝑇 ∈ V → ( mTC ‘ 𝑇 ) = ∅ )
15 1 14 syl5eq ( ¬ 𝑇 ∈ V → 𝐾 = ∅ )
16 15 xpeq1d ( ¬ 𝑇 ∈ V → ( 𝐾 × Word ( 𝐶𝑉 ) ) = ( ∅ × Word ( 𝐶𝑉 ) ) )
17 11 13 16 3eqtr4a ( ¬ 𝑇 ∈ V → 𝐸 = ( 𝐾 × Word ( 𝐶𝑉 ) ) )
18 9 17 pm2.61i 𝐸 = ( 𝐾 × Word ( 𝐶𝑉 ) )