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Mirrors > Home > NFE 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 3972. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4003. Theorem intiin 4021 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 3971 | . 2 class ∩x ∈ A B |
5 | vy | . . . . . 6 setvar y | |
6 | 5 | cv 1641 | . . . . 5 class y |
7 | 6, 3 | wcel 1710 | . . . 4 wff y ∈ B |
8 | 7, 1, 2 | wral 2615 | . . 3 wff ∀x ∈ A y ∈ B |
9 | 8, 5 | cab 2339 | . 2 class {y ∣ ∀x ∈ A y ∈ B} |
10 | 4, 9 | wceq 1642 | 1 wff ∩x ∈ A B = {y ∣ ∀x ∈ A y ∈ B} |
Colors of variables: wff setvar class |
This definition is referenced by: eliin 3975 iineq1 3984 iineq2 3987 nfiin 3997 nfii1 3999 dfiin2g 4001 cbviin 4005 intiin 4021 0iin 4025 viin 4026 iinxsng 4043 |
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