Theorem bj-endval 38330

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 38329	. . 3 ⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
2		fveq2 6886	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		fveq2 6886	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
4	3	oveqd 7438	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
5	4	opeq2d 4846	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩ = ⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩)
6		fveq2 6886	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7	6	oveqd 7438	. . . . . 6 ⊢ (𝑐 = 𝐶 → (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥))
8	7	opeq2d 4846	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩ = ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩)
9	5, 8	preq12d 4698	. . . 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 5191	. . 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 6899	. . . . 5 ⊢ (Base‘𝐶) ∈ V
13	12	mptex 7230	. . . 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 7018	. 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 7439	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥(Hom ‘𝐶)𝑥) = (𝑋(Hom ‘𝐶)𝑋))
18	17	opeq2d 4846	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩ = ⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩)
19	16, 16	opeq12d 4847	. . . . . 6 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
20	19, 16	oveq12d 7439	. . . . 5 ⊢ (𝑥 = 𝑋 → (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
21	20	opeq2d 4846	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩ = ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩)
22	18, 21	preq12d 4698	. . 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 5395	. . 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 7002	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 1569   ∈ wcel 2144  Vcvv 3450  {cpr 4586  ⟨cop 4834   ↦ cmpt 5185  ‘cfv 6539  (class class class)co 7421  ndxcnx 17377  Basecbs 17393  +_gcplusg 17434  Hom chom 17445  compcco 17446  Catccat 17844  End cend 38328

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-bj-end 38329

This theorem is used by:  bj-endbase  38331  bj-endcomp  38332