Theorem isinitoi 16476

Description: Implication of a class being an initial object. (Contributed by AV, 6-Apr-2020.)

Hypotheses

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

Assertion

Ref	Expression
isinitoi	⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏)))

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

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

Proof of Theorem isinitoi

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isinitoi.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		isinitoi.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
3		isinitoi.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
4	1, 2, 3	initoval 16470	. . . . 5 ⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)})
5	4	eleq2d 2673	. . . 4 ⊢ (𝜑 → (𝑂 ∈ (InitO‘𝐶) ↔ 𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)}))
6		elrabi 3328	. . . 4 ⊢ (𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)} → 𝑂 ∈ 𝐵)
7	5, 6	syl6bi 242	. . 3 ⊢ (𝜑 → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ 𝐵))
8	7	imp 444	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → 𝑂 ∈ 𝐵)
9	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat)
10		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵)
11	2, 3, 9, 10	isinito 16473	. . . 4 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏)))
12	11	biimpd 218	. . 3 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (InitO‘𝐶) → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏)))
13	12	impancom 455	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏)))
14	8, 13	jcai 557	1 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  ∀wral 2896  {crab 2900  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  Catccat 16148  InitOcinito 16461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-inito 16464

This theorem is referenced by:  initoid  16478  initoo  16480  initoeu1  16484  initoeu2  16489