Theorem pjch1

Description: Property of identity projection. Remark in Beran p. 111. (Contributed by NM, 28-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	pjch1	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ∈ ℋ ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ ) )
2		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) = ( ( projℎ ‘ ℋ ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
3		id	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
4	2 3	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) = 𝐴 ↔ ( ( projℎ ‘ ℋ ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
5	1 4	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 ∈ ℋ ↔ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) = 𝐴 ) ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ ↔ ( ( projℎ ‘ ℋ ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
6		helch	⊢ ℋ ∈ Cℋ
7		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
8	6 7	pjchi	⊢ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ ↔ ( ( projℎ ‘ ℋ ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
9	5 8	dedth	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 ∈ ℋ ↔ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) = 𝐴 ) )
10	9	ibi	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) = 𝐴 )