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 ) )