Theorem dmin

Description: The domain of an intersection is included in the intersection of the domains. Theorem 6 of Suppes p. 60. (Contributed by NM, 15-Sep-2004)

	Ref	Expression
Assertion	dmin	⊢ dom ( 𝐴 ∩ 𝐵 ) ⊆ ( dom 𝐴 ∩ dom 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		19.40	⊢ ( ∃ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) → ( ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
2		vex	⊢ 𝑥 ∈ V
3	2	eldm2	⊢ ( 𝑥 ∈ dom ( 𝐴 ∩ 𝐵 ) ↔ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 ∩ 𝐵 ) )
4		elin	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
5	4	exbii	⊢ ( ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 ∩ 𝐵 ) ↔ ∃ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
6	3 5	bitri	⊢ ( 𝑥 ∈ dom ( 𝐴 ∩ 𝐵 ) ↔ ∃ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
7		elin	⊢ ( 𝑥 ∈ ( dom 𝐴 ∩ dom 𝐵 ) ↔ ( 𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵 ) )
8	2	eldm2	⊢ ( 𝑥 ∈ dom 𝐴 ↔ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 )
9	2	eldm2	⊢ ( 𝑥 ∈ dom 𝐵 ↔ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 )
10	8 9	anbi12i	⊢ ( ( 𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵 ) ↔ ( ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
11	7 10	bitri	⊢ ( 𝑥 ∈ ( dom 𝐴 ∩ dom 𝐵 ) ↔ ( ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
12	1 6 11	3imtr4i	⊢ ( 𝑥 ∈ dom ( 𝐴 ∩ 𝐵 ) → 𝑥 ∈ ( dom 𝐴 ∩ dom 𝐵 ) )
13	12	ssriv	⊢ dom ( 𝐴 ∩ 𝐵 ) ⊆ ( dom 𝐴 ∩ dom 𝐵 )