Theorem isinito 18048

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 18045	. . 3 ⊢ (𝜑 → (InitO‘𝐶) = {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)})
5	4	eleq2d 2849	. 2 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)}))
6		isinito.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
7		oveq1 7417	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏))
8	7	eleq2d 2849	. . . . . 6 ⊢ (𝑖 = 𝐼 → (ℎ ∈ (𝑖𝐻𝑏) ↔ ℎ ∈ (𝐼𝐻𝑏)))
9	8	eubidv 2614	. . . . 5 ⊢ (𝑖 = 𝐼 → (∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
10	9	ralbidv 3188	. . . 4 ⊢ (𝑖 = 𝐼 → (∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
11	10	elrab3 3651	. . 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 1570   ∈ wcel 2143  ∃!weu 2596  ∀wral 3079  {crab 3416  ‘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:  isinitoi  18051  initoeu2  18068  zrinitorngc  20741  zrninitoringc  20775  irinitoringc  21629  isinito2lem  50276