Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4958. Theorem uniiun 4982 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 7006 and funiunfv 7007 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 4919 | . 2 class ∪ 𝑥 ∈ 𝐴 𝐵 |
5 | vy | . . . . . 6 setvar 𝑦 | |
6 | 5 | cv 1536 | . . . . 5 class 𝑦 |
7 | 6, 3 | wcel 2114 | . . . 4 wff 𝑦 ∈ 𝐵 |
8 | 7, 1, 2 | wrex 3139 | . . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
9 | 8, 5 | cab 2799 | . 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
10 | 4, 9 | wceq 1537 | 1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
Colors of variables: wff setvar class |
This definition is referenced by: eliun 4923 iuneq12df 4945 nfiun 4949 nfiung 4951 nfiu1 4953 dfiunv2 4960 cbviun 4961 cbviung 4963 iunss 4969 uniiun 4982 iunopab 5446 opeliunxp 5619 reliun 5689 fnasrn 6907 abrexex2g 7665 marypha2lem4 8902 cshwsiun 16433 cbviunf 30307 iuneq12daf 30308 iunrdx 30315 iunrnmptss 30317 bnj956 32048 bnj1143 32062 bnj1146 32063 bnj1400 32107 bnj882 32198 bnj18eq1 32199 bnj893 32200 bnj1398 32306 ralssiun 34691 volsupnfl 34952 iunsn 39167 ss2iundf 40053 iunssf 41401 opeliun2xp 44430 nfiund 44826 nfiundg 44827 |
Copyright terms: Public domain | W3C validator |