Theorem isinito 17932

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 17929	. . 3 ⊢ (𝜑 → (InitO‘𝐶) = {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)})
5	4	eleq2d 2823	. 2 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)}))
6		isinito.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
7		oveq1 7375	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏))
8	7	eleq2d 2823	. . . . . 6 ⊢ (𝑖 = 𝐼 → (ℎ ∈ (𝑖𝐻𝑏) ↔ ℎ ∈ (𝐼𝐻𝑏)))
9	8	eubidv 2587	. . . . 5 ⊢ (𝑖 = 𝐼 → (∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
10	9	ralbidv 3161	. . . 4 ⊢ (𝑖 = 𝐼 → (∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
11	10	elrab3 3649	. . 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 1542   ∈ wcel 2114  ∃!weu 2569  ∀wral 3052  {crab 3401  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  Catccat 17599  InitOcinito 17917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-inito 17920

This theorem is referenced by:  isinitoi  17935  initoeu2  17952  zrinitorngc  20587  zrninitoringc  20621  irinitoringc  21446  isinito2lem  49857