![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > df-co | Structured version Visualization version GIF version |
Description: Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 27995) because (cos‘0) = 1 (see cos0 15363) and (exp‘1) = e (see df-e 15282). Note that Definition 7 of [Suppes] p. 63 reverses 𝐴 and 𝐵, uses / instead of ∘, and calls the operation "relative product." (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
df-co | ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cB | . . 3 class 𝐵 | |
3 | 1, 2 | ccom 5411 | . 2 class (𝐴 ∘ 𝐵) |
4 | vx | . . . . . . 7 setvar 𝑥 | |
5 | 4 | cv 1506 | . . . . . 6 class 𝑥 |
6 | vz | . . . . . . 7 setvar 𝑧 | |
7 | 6 | cv 1506 | . . . . . 6 class 𝑧 |
8 | 5, 7, 2 | wbr 4929 | . . . . 5 wff 𝑥𝐵𝑧 |
9 | vy | . . . . . . 7 setvar 𝑦 | |
10 | 9 | cv 1506 | . . . . . 6 class 𝑦 |
11 | 7, 10, 1 | wbr 4929 | . . . . 5 wff 𝑧𝐴𝑦 |
12 | 8, 11 | wa 387 | . . . 4 wff (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) |
13 | 12, 6 | wex 1742 | . . 3 wff ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) |
14 | 13, 4, 9 | copab 4991 | . 2 class {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} |
15 | 3, 14 | wceq 1507 | 1 wff (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} |
Colors of variables: wff setvar class |
This definition is referenced by: coss1 5576 coss2 5577 nfco 5586 brcog 5587 cnvco 5606 cotrg 5811 relco 5936 coundi 5939 coundir 5940 cores 5941 xpco 5978 dffun2 6198 funco 6228 xpcomco 8403 coss12d 14193 xpcogend 14195 trclublem 14216 rtrclreclem3 14280 dfcoss3 35113 |
Copyright terms: Public domain | W3C validator |