Theorem initoid 18226

Description: For an initial object, the identity arrow is the one and only morphism of the object to the object itself. (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
initoid	⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})

Proof of Theorem initoid

Dummy variables ℎ 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isinitoi.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		isinitoi.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		isinitoi.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
4	1, 2, 3	isinitoi 18224	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑜)))
5		oveq2 7459	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑂𝐻𝑜) = (𝑂𝐻𝑂))
6	5	eleq2d 2850	. . . . . . 7 ⊢ (𝑜 = 𝑂 → (ℎ ∈ (𝑂𝐻𝑜) ↔ ℎ ∈ (𝑂𝐻𝑂)))
7	6	eubidv 2614	. . . . . 6 ⊢ (𝑜 = 𝑂 → (∃!ℎ ℎ ∈ (𝑂𝐻𝑜) ↔ ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
8	7	rspcv 3576	. . . . 5 ⊢ (𝑂 ∈ 𝐵 → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑜) → ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
9	8	adantl 487	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑜) → ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
10		eusn 4693	. . . . 5 ⊢ (∃!ℎ ℎ ∈ (𝑂𝐻𝑂) ↔ ∃ℎ(𝑂𝐻𝑂) = {ℎ})
11		eqid 2763	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	3	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat)
13		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵)
14	1, 2, 11, 12, 13	catidcl 17906	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → ((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂))
15		fvex 6929	. . . . . . . . . . . . 13 ⊢ ((Id‘𝐶)‘𝑂) ∈ V
16	15	elsn 4599	. . . . . . . . . . . 12 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} ↔ ((Id‘𝐶)‘𝑂) = ℎ)
17		eqcom 2770	. . . . . . . . . . . 12 ⊢ (((Id‘𝐶)‘𝑂) = ℎ ↔ ℎ = ((Id‘𝐶)‘𝑂))
18		sneqbg 4803	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ V → ({ℎ} = {((Id‘𝐶)‘𝑂)} ↔ ℎ = ((Id‘𝐶)‘𝑂)))
19	18	bicomd 226	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ V → (ℎ = ((Id‘𝐶)‘𝑂) ↔ {ℎ} = {((Id‘𝐶)‘𝑂)}))
20	19	elv 3459	. . . . . . . . . . . 12 ⊢ (ℎ = ((Id‘𝐶)‘𝑂) ↔ {ℎ} = {((Id‘𝐶)‘𝑂)})
21	16, 17, 20	3bitri 300	. . . . . . . . . . 11 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} ↔ {ℎ} = {((Id‘𝐶)‘𝑂)})
22	21	biimpi 219	. . . . . . . . . 10 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} → {ℎ} = {((Id‘𝐶)‘𝑂)})
23	22	a1i 11	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((Id‘𝐶)‘𝑂) ∈ {ℎ} → {ℎ} = {((Id‘𝐶)‘𝑂)}))
24		eleq2 2853	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) ↔ ((Id‘𝐶)‘𝑂) ∈ {ℎ}))
25		eqeq1 2767	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → ((𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)} ↔ {ℎ} = {((Id‘𝐶)‘𝑂)}))
26	23, 24, 25	3imtr4d 297	. . . . . . . 8 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
27	14, 26	syl5 35	. . . . . . 7 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
28	27	exlimiv 1965	. . . . . 6 ⊢ (∃ℎ(𝑂𝐻𝑂) = {ℎ} → (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
29	28	com12 33	. . . . 5 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∃ℎ(𝑂𝐻𝑂) = {ℎ} → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
30	10, 29	biimtrid 245	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∃!ℎ ℎ ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
31	9, 30	syld 48	. . 3 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑜) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
32	31	expimpd 459	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → ((𝑂 ∈ 𝐵 ∧ ∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑜)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
33	4, 32	mpd 16	1 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃!weu 2596  ∀wral 3080  Vcvv 3454  {csn 4588  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  Hom chom 17483  Catccat 17888  Idccid 17889  InitOcinito 18206

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-cat 17892  df-cid 17893  df-inito 18209

This theorem is used by:  2initoinv  18235