Theorem initoval 18045

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 18036	. 2 ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
2		fveq2 6881	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		initoval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eqtr4di 2816	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fveq2 6881	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6		initoval.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶)
7	5, 6	eqtr4di 2816	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
8	7	oveqd 7427	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑎(Hom ‘𝑐)𝑏) = (𝑎𝐻𝑏))
9	8	eleq2d 2849	. . . . 5 ⊢ (𝑐 = 𝐶 → (ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ℎ ∈ (𝑎𝐻𝑏)))
10	9	eubidv 2614	. . . 4 ⊢ (𝑐 = 𝐶 → (∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)))
11	4, 10	raleqbidv 3338	. . 3 ⊢ (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)))
12	4, 11	rabeqbidv 3434	. 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 6895	. . . 4 ⊢ 𝐵 ∈ V
15	14	rabex 5309	. . 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 7013	1 ⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)})

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∃!weu 2596  ∀wral 3079  {crab 3416  Vcvv 3455  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  Hom chom 17316  Catccat 17715  InitOcinito 18033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-inito 18036

This theorem is referenced by:  isinito  18048  isinitoi  18051  dftermo2  18056  initopropdlem  50018  initopropd  50021