Theorem pjhval

Description: Value of a projection. (Contributed by NM, 23-Oct-1999) (Revised by Mario Carneiro, 15-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	pjhval	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pjhfval	⊢ ( 𝐻 ∈ Cℋ → ( projℎ ‘ 𝐻 ) = ( 𝑧 ∈ ℋ ↦ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) ) )
2	1	fveq1d	⊢ ( 𝐻 ∈ Cℋ → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ( 𝑧 ∈ ℋ ↦ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) ) ‘ 𝐴 ) )
3		eqeq1	⊢ ( 𝑧 = 𝐴 → ( 𝑧 = ( 𝑥 +ℎ 𝑦 ) ↔ 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) )
4	3	rexbidv	⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ↔ ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) )
5	4	riotabidv	⊢ ( 𝑧 = 𝐴 → ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) = ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) )
6		eqid	⊢ ( 𝑧 ∈ ℋ ↦ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) ) = ( 𝑧 ∈ ℋ ↦ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) )
7		riotaex	⊢ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) ∈ V
8	5 6 7	fvmpt	⊢ ( 𝐴 ∈ ℋ → ( ( 𝑧 ∈ ℋ ↦ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) ) ‘ 𝐴 ) = ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) )
9	2 8	sylan9eq	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) )