Definition df-of 6061

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) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))

Distinct variable group:   𝑓,𝑔,𝑥,𝑅

Detailed syntax breakdown of Definition df-of

Step	Hyp	Ref	Expression
1		cR	. . 3 class 𝑅
2	1	cof 6059	. 2 class ∘𝑓 𝑅
3		vf	. . 3 setvar 𝑓
4		vg	. . 3 setvar 𝑔
5		cvv 2730	. . 3 class V
6		vx	. . . 4 setvar 𝑥
7	3	cv 1347	. . . . . 6 class 𝑓
8	7	cdm 4611	. . . . 5 class dom 𝑓
9	4	cv 1347	. . . . . 6 class 𝑔
10	9	cdm 4611	. . . . 5 class dom 𝑔
11	8, 10	cin 3120	. . . 4 class (dom 𝑓 ∩ dom 𝑔)
12	6	cv 1347	. . . . . 6 class 𝑥
13	12, 7	cfv 5198	. . . . 5 class (𝑓‘𝑥)
14	12, 9	cfv 5198	. . . . 5 class (𝑔‘𝑥)
15	13, 14, 1	co 5853	. . . 4 class ((𝑓‘𝑥)𝑅(𝑔‘𝑥))
16	6, 11, 15	cmpt 4050	. . 3 class (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))
17	3, 4, 5, 5, 16	cmpo 5855	. 2 class (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
18	2, 17	wceq 1348	1 wff ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))

This definition is referenced by:  ofeq  6063  ofexg  6065  nfof  6066  offval  6068  offval3  6113  ofmres  6115