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Definition df-cofu 16291
Description: Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
Assertion
Ref Expression
df-cofu func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st𝑔) ∘ (1st𝑓)), (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))⟩)
Distinct variable group:   𝑓,𝑔,𝑥,𝑦

Detailed syntax breakdown of Definition df-cofu
StepHypRef Expression
1 ccofu 16287 . 2 class func
2 vg . . 3 setvar 𝑔
3 vf . . 3 setvar 𝑓
4 cvv 3172 . . 3 class V
52cv 1473 . . . . . 6 class 𝑔
6 c1st 7034 . . . . . 6 class 1st
75, 6cfv 5789 . . . . 5 class (1st𝑔)
83cv 1473 . . . . . 6 class 𝑓
98, 6cfv 5789 . . . . 5 class (1st𝑓)
107, 9ccom 5031 . . . 4 class ((1st𝑔) ∘ (1st𝑓))
11 vx . . . . 5 setvar 𝑥
12 vy . . . . 5 setvar 𝑦
13 c2nd 7035 . . . . . . . 8 class 2nd
148, 13cfv 5789 . . . . . . 7 class (2nd𝑓)
1514cdm 5027 . . . . . 6 class dom (2nd𝑓)
1615cdm 5027 . . . . 5 class dom dom (2nd𝑓)
1711cv 1473 . . . . . . . 8 class 𝑥
1817, 9cfv 5789 . . . . . . 7 class ((1st𝑓)‘𝑥)
1912cv 1473 . . . . . . . 8 class 𝑦
2019, 9cfv 5789 . . . . . . 7 class ((1st𝑓)‘𝑦)
215, 13cfv 5789 . . . . . . 7 class (2nd𝑔)
2218, 20, 21co 6526 . . . . . 6 class (((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦))
2317, 19, 14co 6526 . . . . . 6 class (𝑥(2nd𝑓)𝑦)
2422, 23ccom 5031 . . . . 5 class ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦))
2511, 12, 16, 16, 24cmpt2 6528 . . . 4 class (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))
2610, 25cop 4130 . . 3 class ⟨((1st𝑔) ∘ (1st𝑓)), (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))⟩
272, 3, 4, 4, 26cmpt2 6528 . 2 class (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st𝑔) ∘ (1st𝑓)), (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))⟩)
281, 27wceq 1474 1 wff func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st𝑔) ∘ (1st𝑓)), (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))⟩)
Colors of variables: wff setvar class
This definition is referenced by:  cofuval  16313
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