Theorem bj-endval 37355

Description: Value of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-endval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
bj-endval.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))

Assertion

Ref	Expression
bj-endval	⊢ (𝜑 → ((End ‘𝐶)‘𝑋) = {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩})

Proof of Theorem bj-endval

Dummy variables 𝑐 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-end 37354	. . 3 ⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
2		fveq2 6822	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		fveq2 6822	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
4	3	oveqd 7363	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
5	4	opeq2d 4832	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩ = ⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩)
6		fveq2 6822	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7	6	oveqd 7363	. . . . . 6 ⊢ (𝑐 = 𝐶 → (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥))
8	7	opeq2d 4832	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩ = ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩)
9	5, 8	preq12d 4694	. . . 4 ⊢ (𝑐 = 𝐶 → {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩} = {⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩})
10	2, 9	mpteq12dv 5178	. . 3 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}) = (𝑥 ∈ (Base‘𝐶) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩}))
11		bj-endval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
12		fvex 6835	. . . . 5 ⊢ (Base‘𝐶) ∈ V
13	12	mptex 7157	. . . 4 ⊢ (𝑥 ∈ (Base‘𝐶) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩}) ∈ V
14	13	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩}) ∈ V)
15	1, 10, 11, 14	fvmptd3 6952	. 2 ⊢ (𝜑 → (End ‘𝐶) = (𝑥 ∈ (Base‘𝐶) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩}))
16		id 22	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
17	16, 16	oveq12d 7364	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥(Hom ‘𝐶)𝑥) = (𝑋(Hom ‘𝐶)𝑋))
18	17	opeq2d 4832	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩ = ⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩)
19	16, 16	opeq12d 4833	. . . . . 6 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
20	19, 16	oveq12d 7364	. . . . 5 ⊢ (𝑥 = 𝑋 → (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
21	20	opeq2d 4832	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩ = ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩)
22	18, 21	preq12d 4694	. . 3 ⊢ (𝑥 = 𝑋 → {⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩} = {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩})
23	22	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → {⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩} = {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩})
24		bj-endval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
25		prex 5375	. . 3 ⊢ {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩} ∈ V
26	25	a1i 11	. 2 ⊢ (𝜑 → {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩} ∈ V)
27	15, 23, 24, 26	fvmptd 6936	1 ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) = {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436  {cpr 4578  ⟨cop 4582   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346  ndxcnx 17104  Basecbs 17120  +_gcplusg 17161  Hom chom 17172  compcco 17173  Catccat 17570  End cend 37353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-bj-end 37354

This theorem is referenced by:  bj-endbase  37356  bj-endcomp  37357