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Mirrors > Home > MPE Home > Th. List > df-iin | Structured version Visualization version GIF version |
Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4997. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 5038. Theorem intiin 5063 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 4996 | . 2 class ∩ 𝑥 ∈ 𝐴 𝐵 |
5 | vy | . . . . . 6 setvar 𝑦 | |
6 | 5 | cv 1535 | . . . . 5 class 𝑦 |
7 | 6, 3 | wcel 2105 | . . . 4 wff 𝑦 ∈ 𝐵 |
8 | 7, 1, 2 | wral 3058 | . . 3 wff ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
9 | 8, 5 | cab 2711 | . 2 class {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
10 | 4, 9 | wceq 1536 | 1 wff ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
Colors of variables: wff setvar class |
This definition is referenced by: eliin 5000 iineq1 5013 iineq2 5016 nfiin 5028 nfiing 5030 nfii1 5033 dfiin2g 5036 cbviin 5041 cbviing 5043 cbviinv 5045 intiin 5063 0iin 5068 viin 5069 iinxsng 5092 iinxprg 5093 iinuni 5102 iinabrex 32588 iineq1i 36177 iineq12i 36178 cbviinvw2 36215 cbviindavw 36245 cbviindavw2 36269 iineq12f 38150 iineq12dv 45045 |
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