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	$$⊢S=\left(x\in {\mathbf{C}}_{\mathscr{H}}⟼{\mathrm{proj}}_{ℎ}\left(x\right)\left(u\right)\right)$$
Assertion	hstrlem2	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}\to S\left(C\right)={\mathrm{proj}}_{ℎ}\left(C\right)\left(u\right)$$

Proof

Step	Hyp	Ref	Expression
1		hstrlem2.1	$$⊢S=\left(x\in {\mathbf{C}}_{\mathscr{H}}⟼{\mathrm{proj}}_{ℎ}\left(x\right)\left(u\right)\right)$$
2		fveq2	$$⊢x=C\to {\mathrm{proj}}_{ℎ}\left(x\right)={\mathrm{proj}}_{ℎ}\left(C\right)$$
3	2	fveq1d	$$⊢x=C\to {\mathrm{proj}}_{ℎ}\left(x\right)\left(u\right)={\mathrm{proj}}_{ℎ}\left(C\right)\left(u\right)$$
4		fvex	$$⊢{\mathrm{proj}}_{ℎ}\left(C\right)\left(u\right)\in \mathrm{V}$$
5	3 1 4	fvmpt	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}\to S\left(C\right)={\mathrm{proj}}_{ℎ}\left(C\right)\left(u\right)$$