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Mirrors > Home > MPE Home > Th. List > df-iun | Structured version Visualization version GIF version |
Description: Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, 𝐴 is independent of 𝑥 (although this is not required by the definition), and 𝐵 depends on 𝑥 i.e. can be read informally as 𝐵(𝑥). We call 𝑥 the index, 𝐴 the index set, and 𝐵 the indexed set. In most books, 𝑥 ∈ 𝐴 is written as a subscript or underneath a union symbol ∪. We use a special union symbol ∪ to make it easier to distinguish from plain class union. In many theorems, you will see that 𝑥 and 𝐴 are in the same distinct variable group (meaning 𝐴 cannot depend on 𝑥) and that 𝐵 and 𝑥 do not share a distinct variable group (meaning that can be thought of as 𝐵(𝑥) i.e. can be substituted with a class expression containing 𝑥). An alternate definition tying indexed union to ordinary union is dfiun2 4586. Theorem uniiun 4605 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6545 and funiunfv 6546 are useful when 𝐵 is a function. (Contributed by NM, 27-Jun-1998.) |
Ref | Expression |
---|---|
df-iun | ⊢ ∪ 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 | ciun 4552 | . 2 class ∪ 𝑥 ∈ 𝐴 𝐵 |
5 | vy | . . . . . 6 setvar 𝑦 | |
6 | 5 | cv 1522 | . . . . 5 class 𝑦 |
7 | 6, 3 | wcel 2030 | . . . 4 wff 𝑦 ∈ 𝐵 |
8 | 7, 1, 2 | wrex 2942 | . . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
9 | 8, 5 | cab 2637 | . 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
10 | 4, 9 | wceq 1523 | 1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
Colors of variables: wff setvar class |
This definition is referenced by: eliun 4556 iuneq12df 4576 nfiun 4580 nfiu1 4582 dfiunv2 4588 cbviun 4589 iunss 4593 uniiun 4605 iunopab 5041 opeliunxp 5204 reliun 5272 fnasrn 6451 abrexex2g 7186 abrexex2OLD 7192 marypha2lem4 8385 cshwsiun 15853 cbviunf 29498 iuneq12daf 29499 iunrdx 29508 bnj956 30973 bnj1143 30987 bnj1146 30988 bnj1400 31032 bnj882 31122 bnj18eq1 31123 bnj893 31124 bnj1398 31228 volsupnfl 33584 ss2iundf 38268 iunssf 39577 opeliun2xp 42436 nfiund 42746 |
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