Theorem bj-endval 37382

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 37381	. . 3 ⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
2		fveq2 6830	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		fveq2 6830	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
4	3	oveqd 7371	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
5	4	opeq2d 4833	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩ = ⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩)
6		fveq2 6830	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7	6	oveqd 7371	. . . . . 6 ⊢ (𝑐 = 𝐶 → (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥))
8	7	opeq2d 4833	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩ = ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩)
9	5, 8	preq12d 4695	. . . 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 5182	. . 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 6843	. . . . 5 ⊢ (Base‘𝐶) ∈ V
13	12	mptex 7165	. . . 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 6960	. 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 7372	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥(Hom ‘𝐶)𝑥) = (𝑋(Hom ‘𝐶)𝑋))
18	17	opeq2d 4833	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩ = ⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩)
19	16, 16	opeq12d 4834	. . . . . 6 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
20	19, 16	oveq12d 7372	. . . . 5 ⊢ (𝑥 = 𝑋 → (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
21	20	opeq2d 4833	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩ = ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩)
22	18, 21	preq12d 4695	. . 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 5379	. . 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 6944	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 2113  Vcvv 3437  {cpr 4579  ⟨cop 4583   ↦ cmpt 5176  ‘cfv 6488  (class class class)co 7354  ndxcnx 17108  Basecbs 17124  +_gcplusg 17165  Hom chom 17176  compcco 17177  Catccat 17574  End cend 37380

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7357  df-bj-end 37381

This theorem is referenced by:  bj-endbase  37383  bj-endcomp  37384