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Metamath Proof Explorer

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 ) ) )