Theorem bj-endval 37629

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 37628	. . 3 ⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
2		fveq2 6840	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		fveq2 6840	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
4	3	oveqd 7384	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
5	4	opeq2d 4823	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩ = ⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩)
6		fveq2 6840	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7	6	oveqd 7384	. . . . . 6 ⊢ (𝑐 = 𝐶 → (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥))
8	7	opeq2d 4823	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩ = ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩)
9	5, 8	preq12d 4685	. . . 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 5172	. . 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 6853	. . . . 5 ⊢ (Base‘𝐶) ∈ V
13	12	mptex 7178	. . . 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 6971	. 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 7385	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥(Hom ‘𝐶)𝑥) = (𝑋(Hom ‘𝐶)𝑋))
18	17	opeq2d 4823	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(Base‘ndx), (𝑥(Hom ‘𝐶)𝑥)⟩ = ⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩)
19	16, 16	opeq12d 4824	. . . . . 6 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
20	19, 16	oveq12d 7385	. . . . 5 ⊢ (𝑥 = 𝑋 → (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
21	20	opeq2d 4823	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑥)⟩ = ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩)
22	18, 21	preq12d 4685	. . 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 5380	. . 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 6955	1 ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) = {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429  {cpr 4569  ⟨cop 4573   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367  ndxcnx 17163  Basecbs 17179  +_gcplusg 17220  Hom chom 17231  compcco 17232  Catccat 17630  End cend 37627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-bj-end 37628

This theorem is referenced by:  bj-endbase  37630  bj-endcomp  37631