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Mirrors > Home > ILE Home > Th. List > df-iin | GIF version |
Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3815. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3848. Theorem intiin 3867 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.) |
Ref | Expression |
---|---|
df-iin | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . 3 setvar 𝑥 | |
2 | cA | . . 3 class 𝐴 | |
3 | cB | . . 3 class 𝐵 | |
4 | 1, 2, 3 | ciin 3814 | . 2 class ∩ 𝑥 ∈ 𝐴 𝐵 |
5 | vy | . . . . . 6 setvar 𝑦 | |
6 | 5 | cv 1330 | . . . . 5 class 𝑦 |
7 | 6, 3 | wcel 1480 | . . . 4 wff 𝑦 ∈ 𝐵 |
8 | 7, 1, 2 | wral 2416 | . . 3 wff ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
9 | 8, 5 | cab 2125 | . 2 class {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
10 | 4, 9 | wceq 1331 | 1 wff ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
Colors of variables: wff set class |
This definition is referenced by: eliin 3818 iineq1 3827 iineq2 3830 nfiinxy 3840 nfiinya 3842 nfii1 3844 dfiin2g 3846 cbviin 3851 intiin 3867 0iin 3871 viin 3872 iinxsng 3886 iinxprg 3887 iinuniss 3895 bdciin 13077 |
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