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Definition df-of 5654
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 5652 . 2 class 𝑓 𝑅
3 vf . . 3 setvar f
4 vg . . 3 setvar g
5 cvv 2551 . . 3 class V
6 vx . . . 4 setvar x
73cv 1241 . . . . . 6 class f
87cdm 4288 . . . . 5 class dom f
94cv 1241 . . . . . 6 class g
109cdm 4288 . . . . 5 class dom g
118, 10cin 2910 . . . 4 class (dom f ∩ dom g)
126cv 1241 . . . . . 6 class x
1312, 7cfv 4845 . . . . 5 class (fx)
1412, 9cfv 4845 . . . . 5 class (gx)
1513, 14, 1co 5455 . . . 4 class ((fx)𝑅(gx))
166, 11, 15cmpt 3809 . . 3 class (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx)))
173, 4, 5, 5, 16cmpt2 5457 . 2 class (f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))))
182, 17wceq 1242 1 wff 𝑓 𝑅 = (f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))))
Colors of variables: wff set class
This definition is referenced by:  ofeq  5656  ofexg  5658  offval  5661  offval3  5703  ofmres  5705
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