Theorem hstrlem2

Description: Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	hstrlem2.1	⊢ 𝑆 = ( 𝑥 ∈ Cℋ ↦ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) )
Assertion	hstrlem2	⊢ ( 𝐶 ∈ Cℋ → ( 𝑆 ‘ 𝐶 ) = ( ( projℎ ‘ 𝐶 ) ‘ 𝑢 ) )

Proof

Step	Hyp	Ref	Expression
1		hstrlem2.1	⊢ 𝑆 = ( 𝑥 ∈ Cℋ ↦ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) )
2		fveq2	⊢ ( 𝑥 = 𝐶 → ( projℎ ‘ 𝑥 ) = ( projℎ ‘ 𝐶 ) )
3	2	fveq1d	⊢ ( 𝑥 = 𝐶 → ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) = ( ( projℎ ‘ 𝐶 ) ‘ 𝑢 ) )
4		fvex	⊢ ( ( projℎ ‘ 𝐶 ) ‘ 𝑢 ) ∈ V
5	3 1 4	fvmpt	⊢ ( 𝐶 ∈ Cℋ → ( 𝑆 ‘ 𝐶 ) = ( ( projℎ ‘ 𝐶 ) ‘ 𝑢 ) )