Theorem bj-endval 38441

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 38440	. . 3 ⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
2		fveq2 6911	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		fveq2 6911	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
4	3	oveqd 7463	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
5	4	opeq2d 4850	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩ = ⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩)
6		fveq2 6911	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7	6	oveqd 7463	. . . . . 6 ⊢ (𝑐 = 𝐶 → (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥))
8	7	opeq2d 4850	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩ = ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩)
9	5, 8	preq12d 4702	. . . 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 5196	. . 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 6924	. . . . 5 ⊢ (Base‘𝐶) ∈ V
13	12	mptex 7255	. . . 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 7043	. 2 ⊢ (𝜑 → (End ‘𝐶) = (𝑥 ∈ (Base‘𝐶) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩}))
16		id 23	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
17	16, 16	oveq12d 7464	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥(Hom ‘𝐶)𝑥) = (𝑋(Hom ‘𝐶)𝑋))
18	17	opeq2d 4850	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩ = ⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩)
19	16, 16	opeq12d 4851	. . . . . 6 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
20	19, 16	oveq12d 7464	. . . . 5 ⊢ (𝑥 = 𝑋 → (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
21	20	opeq2d 4850	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩ = ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩)
22	18, 21	preq12d 4702	. . 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 487	. 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 5401	. . 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 7027	1 ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) = {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {cpr 4590  ⟨cop 4838   ↦ cmpt 5190  ‘cfv 6561  (class class class)co 7446  ndxcnx 17403  Basecbs 17419  +_gcplusg 17460  Hom chom 17471  compcco 17472  Catccat 17870  End cend 38439

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-bj-end 38440

This theorem is used by:  bj-endbase  38442  bj-endcomp  38443