Theorem initoval 17757

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 17748	. 2 ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
2		fveq2 6804	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		initoval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eqtr4di 2794	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fveq2 6804	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6		initoval.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶)
7	5, 6	eqtr4di 2794	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
8	7	oveqd 7324	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑎(Hom ‘𝑐)𝑏) = (𝑎𝐻𝑏))
9	8	eleq2d 2822	. . . . 5 ⊢ (𝑐 = 𝐶 → (ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ℎ ∈ (𝑎𝐻𝑏)))
10	9	eubidv 2584	. . . 4 ⊢ (𝑐 = 𝐶 → (∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)))
11	4, 10	raleqbidv 3314	. . 3 ⊢ (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)))
12	4, 11	rabeqbidv 3427	. 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 6818	. . . 4 ⊢ 𝐵 ∈ V
15	14	rabex 5265	. . 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 6930	1 ⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  ∃!weu 2566  ∀wral 3061  {crab 3330  Vcvv 3437  ‘cfv 6458  (class class class)co 7307  Basecbs 16961  Hom chom 17022  Catccat 17422  InitOcinito 17745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3333  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-iota 6410  df-fun 6460  df-fv 6466  df-ov 7310  df-inito 17748

This theorem is referenced by:  isinito  17760  isinitoi  17763  dftermo2  17768