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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 26687) because (cos‘0) = 1 (see cos0 14719) and (exp‘1) = e (see df-e 14638). 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 5042 | . 2 class (𝐴 ∘ 𝐵) |
4 | vx | . . . . . . 7 setvar 𝑥 | |
5 | 4 | cv 1474 | . . . . . 6 class 𝑥 |
6 | vz | . . . . . . 7 setvar 𝑧 | |
7 | 6 | cv 1474 | . . . . . 6 class 𝑧 |
8 | 5, 7, 2 | wbr 4583 | . . . . 5 wff 𝑥𝐵𝑧 |
9 | vy | . . . . . . 7 setvar 𝑦 | |
10 | 9 | cv 1474 | . . . . . 6 class 𝑦 |
11 | 7, 10, 1 | wbr 4583 | . . . . 5 wff 𝑧𝐴𝑦 |
12 | 8, 11 | wa 383 | . . . 4 wff (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) |
13 | 12, 6 | wex 1695 | . . 3 wff ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) |
14 | 13, 4, 9 | copab 4642 | . 2 class {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} |
15 | 3, 14 | wceq 1475 | 1 wff (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} |
Colors of variables: wff setvar class |
This definition is referenced by: coss1 5199 coss2 5200 nfco 5209 brcog 5210 cnvco 5230 cotrg 5426 relco 5550 coundi 5553 coundir 5554 cores 5555 xpco 5592 dffun2 5814 funco 5842 xpcomco 7935 coss12d 13559 xpcogend 13561 trclublem 13582 rtrclreclem3 13648 |
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