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Definition df-bj-end 36281
Description: The monoid of endomorphisms on an object of a category. (Contributed by BJ, 4-Apr-2024.)
Assertion
Ref Expression
df-bj-end End = (𝑐 ∈ Cat ↦ (π‘₯ ∈ (Baseβ€˜π‘) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩}))
Distinct variable group:   π‘₯,𝑐

Detailed syntax breakdown of Definition df-bj-end
StepHypRef Expression
1 cend 36280 . 2 class End
2 vc . . 3 setvar 𝑐
3 ccat 17610 . . 3 class Cat
4 vx . . . 4 setvar π‘₯
52cv 1540 . . . . 5 class 𝑐
6 cbs 17146 . . . . 5 class Base
75, 6cfv 6543 . . . 4 class (Baseβ€˜π‘)
8 cnx 17128 . . . . . . 7 class ndx
98, 6cfv 6543 . . . . . 6 class (Baseβ€˜ndx)
104cv 1540 . . . . . . 7 class π‘₯
11 chom 17210 . . . . . . . 8 class Hom
125, 11cfv 6543 . . . . . . 7 class (Hom β€˜π‘)
1310, 10, 12co 7411 . . . . . 6 class (π‘₯(Hom β€˜π‘)π‘₯)
149, 13cop 4634 . . . . 5 class ⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩
15 cplusg 17199 . . . . . . 7 class +g
168, 15cfv 6543 . . . . . 6 class (+gβ€˜ndx)
1710, 10cop 4634 . . . . . . 7 class ⟨π‘₯, π‘₯⟩
18 cco 17211 . . . . . . . 8 class comp
195, 18cfv 6543 . . . . . . 7 class (compβ€˜π‘)
2017, 10, 19co 7411 . . . . . 6 class (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)
2116, 20cop 4634 . . . . 5 class ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩
2214, 21cpr 4630 . . . 4 class {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩}
234, 7, 22cmpt 5231 . . 3 class (π‘₯ ∈ (Baseβ€˜π‘) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩})
242, 3, 23cmpt 5231 . 2 class (𝑐 ∈ Cat ↦ (π‘₯ ∈ (Baseβ€˜π‘) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩}))
251, 24wceq 1541 1 wff End = (𝑐 ∈ Cat ↦ (π‘₯ ∈ (Baseβ€˜π‘) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩}))
Colors of variables: wff setvar class
This definition is referenced by:  bj-endval  36282
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