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Metamath Proof Explorer

Theorem pjhfo

Description: A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006) (New usage is discouraged.)

Ref Expression
Assertion pjhfo ( 𝐻 ∈ Cℋ → ( projℎ ‘ 𝐻 ) : ℋ –onto→ 𝐻 )

Proof

Step Hyp Ref Expression
1 fveq2 ⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( projℎ ‘ 𝐻 ) = ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) )
2 foeq1 ⊢ ( ( projℎ ‘ 𝐻 ) = ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) → ( ( projℎ ‘ 𝐻 ) : ℋ –onto→ 𝐻 ↔ ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) : ℋ –onto→ 𝐻 ) )
3 1 2 syl ⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( ( projℎ ‘ 𝐻 ) : ℋ –onto→ 𝐻 ↔ ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) : ℋ –onto→ 𝐻 ) )
4 foeq3 ⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) : ℋ –onto→ 𝐻 ↔ ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) : ℋ –onto→ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) )
5 h0elch ⊢ 0ℋ ∈ Cℋ
6 5 elimel ⊢ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ∈ Cℋ
7 6 pjfoi ⊢ ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) : ℋ –onto→ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ )
8 3 4 7 dedth2v ⊢ ( 𝐻 ∈ Cℋ → ( projℎ ‘ 𝐻 ) : ℋ –onto→ 𝐻 )