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Metamath Proof Explorer

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 )