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Definition df-of 6768
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.)
Assertion
Ref Expression
df-of 𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
Distinct variable group:   𝑓,𝑔,𝑥,𝑅

Detailed syntax breakdown of Definition df-of
StepHypRef Expression
1 cR . . 3 class 𝑅
21cof 6766 . 2 class 𝑓 𝑅
3 vf . . 3 setvar 𝑓
4 vg . . 3 setvar 𝑔
5 cvv 3168 . . 3 class V
6 vx . . . 4 setvar 𝑥
73cv 1473 . . . . . 6 class 𝑓
87cdm 5024 . . . . 5 class dom 𝑓
94cv 1473 . . . . . 6 class 𝑔
109cdm 5024 . . . . 5 class dom 𝑔
118, 10cin 3534 . . . 4 class (dom 𝑓 ∩ dom 𝑔)
126cv 1473 . . . . . 6 class 𝑥
1312, 7cfv 5786 . . . . 5 class (𝑓𝑥)
1412, 9cfv 5786 . . . . 5 class (𝑔𝑥)
1513, 14, 1co 6523 . . . 4 class ((𝑓𝑥)𝑅(𝑔𝑥))
166, 11, 15cmpt 4633 . . 3 class (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))
173, 4, 5, 5, 16cmpt2 6525 . 2 class (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
182, 17wceq 1474 1 wff 𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
Colors of variables: wff setvar class
This definition is referenced by:  ofeq  6770  ofexg  6772  offval  6775  offval3  7026  ofmres  7028  offval0  42091
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