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Metamath Proof Explorer

Theorem dfiin2

Description: Alternate definition of indexed intersection when B is a set. Definition 15(b) of Suppes p. 44. (Contributed by NM, 28-Jun-1998) (Proof shortened by Andrew Salmon, 25-Jul-2011)

		Ref	Expression
	Hypothesis	dfiun2.1	⊢ 𝐵 ∈ V
	Assertion	dfiin2	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 }

Proof

Step	Hyp	Ref	Expression
1		dfiun2.1	⊢ 𝐵 ∈ V
2		dfiin2g	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } )
3	1	a1i	⊢ ( 𝑥 ∈ 𝐴 → 𝐵 ∈ V )
4	2 3	mprg	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 }