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Metamath Proof Explorer

Theorem resmpo

Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013)

Ref Expression
Assertion resmpo ( ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐸 ) ↾ ( 𝐶 × 𝐷 ) ) = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝐸 ) )

Proof

Step Hyp Ref Expression
1 resoprab2 ⊢ ( ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) → ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐸 ) } ↾ ( 𝐶 × 𝐷 ) ) = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 = 𝐸 ) } )
2 df-mpo ⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐸 ) = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐸 ) }
3 2 reseq1i ⊢ ( ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐸 ) ↾ ( 𝐶 × 𝐷 ) ) = ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 𝐸 ) } ↾ ( 𝐶 × 𝐷 ) )
4 df-mpo ⊢ ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝐸 ) = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 = 𝐸 ) }
5 1 3 4 3eqtr4g ⊢ ( ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐸 ) ↾ ( 𝐶 × 𝐷 ) ) = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝐸 ) )