Theorem isinito 18049

Description: The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.)

Hypotheses

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

Assertion

Ref	Expression
isinito	⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))

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

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

Proof of Theorem isinito

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isinito.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		isinito.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		isinito.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
4	1, 2, 3	initoval 18046	. . 3 ⊢ (𝜑 → (InitO‘𝐶) = {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)})
5	4	eleq2d 2855	. 2 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)}))
6		isinito.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
7		oveq1 7415	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏))
8	7	eleq2d 2855	. . . . . 6 ⊢ (𝑖 = 𝐼 → (ℎ ∈ (𝑖𝐻𝑏) ↔ ℎ ∈ (𝐼𝐻𝑏)))
9	8	eubidv 2620	. . . . 5 ⊢ (𝑖 = 𝐼 → (∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
10	9	ralbidv 3194	. . . 4 ⊢ (𝑖 = 𝐼 → (∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
11	10	elrab3 3660	. . 3 ⊢ (𝐼 ∈ 𝐵 → (𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
12	6, 11	syl 18	. 2 ⊢ (𝜑 → (𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
13	5, 12	bitrd 282	1 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∃!weu 2602  ∀wral 3085  {crab 3423  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  Hom chom 17317  Catccat 17716  InitOcinito 18034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fv 6542  df-ov 7411  df-inito 18037

This theorem is referenced by:  isinitoi  18052  initoeu2  18069  zrinitorngc  20723  zrninitoringc  20757  irinitoringc  21594  isinito2lem  50156