Theorem mzpval

Description: Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014)

	Ref	Expression
Assertion	mzpval	⊢ ( 𝑉 ∈ V → ( mzPoly ‘ 𝑉 ) = ∩ ( mzPolyCld ‘ 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		mzpcln0	⊢ ( 𝑉 ∈ V → ( mzPolyCld ‘ 𝑉 ) ≠ ∅ )
2		intex	⊢ ( ( mzPolyCld ‘ 𝑉 ) ≠ ∅ ↔ ∩ ( mzPolyCld ‘ 𝑉 ) ∈ V )
3	1 2	sylib	⊢ ( 𝑉 ∈ V → ∩ ( mzPolyCld ‘ 𝑉 ) ∈ V )
4		fveq2	⊢ ( 𝑣 = 𝑉 → ( mzPolyCld ‘ 𝑣 ) = ( mzPolyCld ‘ 𝑉 ) )
5	4	inteqd	⊢ ( 𝑣 = 𝑉 → ∩ ( mzPolyCld ‘ 𝑣 ) = ∩ ( mzPolyCld ‘ 𝑉 ) )
6		df-mzp	⊢ mzPoly = ( 𝑣 ∈ V ↦ ∩ ( mzPolyCld ‘ 𝑣 ) )
7	5 6	fvmptg	⊢ ( ( 𝑉 ∈ V ∧ ∩ ( mzPolyCld ‘ 𝑉 ) ∈ V ) → ( mzPoly ‘ 𝑉 ) = ∩ ( mzPolyCld ‘ 𝑉 ) )
8	3 7	mpdan	⊢ ( 𝑉 ∈ V → ( mzPoly ‘ 𝑉 ) = ∩ ( mzPolyCld ‘ 𝑉 ) )