Theorem isinito 17922

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 17919	. . 3 ⊢ (𝜑 → (InitO‘𝐶) = {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)})
5	4	eleq2d 2814	. 2 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)}))
6		isinito.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
7		oveq1 7360	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏))
8	7	eleq2d 2814	. . . . . 6 ⊢ (𝑖 = 𝐼 → (ℎ ∈ (𝑖𝐻𝑏) ↔ ℎ ∈ (𝐼𝐻𝑏)))
9	8	eubidv 2579	. . . . 5 ⊢ (𝑖 = 𝐼 → (∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
10	9	ralbidv 3152	. . . 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 17	. 2 ⊢ (𝜑 → (𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
13	5, 12	bitrd 279	1 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∃!weu 2561  ∀wral 3044  {crab 3396  ‘cfv 6486  (class class class)co 7353  Basecbs 17139  Hom chom 17191  Catccat 17589  InitOcinito 17907

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-inito 17910

This theorem is referenced by:  isinitoi  17925  initoeu2  17942  zrinitorngc  20546  zrninitoringc  20580  irinitoringc  21405  isinito2lem  49503