Theorem initoval 18068

Description: The value of the initial object function, i.e. the set of all initial objects of a category. (Contributed by AV, 3-Apr-2020.)

Hypotheses

Ref	Expression
initoval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
initoval.b	⊢ 𝐵 = (Base‘𝐶)
initoval.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
initoval	⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)})

Distinct variable groups:   𝑎,𝑏,ℎ   𝐵,𝑎,𝑏   𝐶,𝑎,𝑏,ℎ

Allowed substitution hints:   𝜑(ℎ, 𝑎, 𝑏)   𝐵(ℎ)   𝐻(ℎ, 𝑎, 𝑏)

Proof of Theorem initoval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inito 18059	. 2 ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
2		fveq2 6885	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		initoval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eqtr4di 2818	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fveq2 6885	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6		initoval.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶)
7	5, 6	eqtr4di 2818	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
8	7	oveqd 7433	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑎(Hom ‘𝑐)𝑏) = (𝑎𝐻𝑏))
9	8	eleq2d 2851	. . . . 5 ⊢ (𝑐 = 𝐶 → (ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ℎ ∈ (𝑎𝐻𝑏)))
10	9	eubidv 2616	. . . 4 ⊢ (𝑐 = 𝐶 → (∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)))
11	4, 10	raleqbidv 3340	. . 3 ⊢ (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)))
12	4, 11	rabeqbidv 3436	. 2 ⊢ (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)} = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)})
13		initoval.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	3	fvexi 6899	. . . 4 ⊢ 𝐵 ∈ V
15	14	rabex 5311	. . 3 ⊢ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)} ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)} ∈ V)
17	1, 12, 13, 16	fvmptd3 7017	1 ⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ∃!weu 2598  ∀wral 3081  {crab 3418  Vcvv 3457  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  Hom chom 17339  Catccat 17738  InitOcinito 18056

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-inito 18059

This theorem is used by:  isinito  18071  isinitoi  18074  dftermo2  18079  initopropdlem  50051  initopropd  50054