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Mirrors > Home > MPE Home > Th. List > df-of | Structured version Visualization version GIF version |
Description: Define the function operation map. The definition is designed so that if 𝑅 is a binary operation, then ∘f 𝑅 is the analogous operation on functions which corresponds to applying 𝑅 pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
df-of | ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cR | . . 3 class 𝑅 | |
2 | 1 | cof 7540 | . 2 class ∘f 𝑅 |
3 | vf | . . 3 setvar 𝑓 | |
4 | vg | . . 3 setvar 𝑔 | |
5 | cvv 3433 | . . 3 class V | |
6 | vx | . . . 4 setvar 𝑥 | |
7 | 3 | cv 1538 | . . . . . 6 class 𝑓 |
8 | 7 | cdm 5590 | . . . . 5 class dom 𝑓 |
9 | 4 | cv 1538 | . . . . . 6 class 𝑔 |
10 | 9 | cdm 5590 | . . . . 5 class dom 𝑔 |
11 | 8, 10 | cin 3887 | . . . 4 class (dom 𝑓 ∩ dom 𝑔) |
12 | 6 | cv 1538 | . . . . . 6 class 𝑥 |
13 | 12, 7 | cfv 6437 | . . . . 5 class (𝑓‘𝑥) |
14 | 12, 9 | cfv 6437 | . . . . 5 class (𝑔‘𝑥) |
15 | 13, 14, 1 | co 7284 | . . . 4 class ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) |
16 | 6, 11, 15 | cmpt 5158 | . . 3 class (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) |
17 | 3, 4, 5, 5, 16 | cmpo 7286 | . 2 class (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) |
18 | 2, 17 | wceq 1539 | 1 wff ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) |
Colors of variables: wff setvar class |
This definition is referenced by: ofeqd 7544 ofexg 7547 nfof 7548 offval 7551 offval3 7834 ofmres 7836 |
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