Theorem initoid 18017

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 18015	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑜)))
5		oveq2 7400	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑂𝐻𝑜) = (𝑂𝐻𝑂))
6	5	eleq2d 2847	. . . . . . 7 ⊢ (𝑜 = 𝑂 → (ℎ ∈ (𝑂𝐻𝑜) ↔ ℎ ∈ (𝑂𝐻𝑂)))
7	6	eubidv 2612	. . . . . 6 ⊢ (𝑜 = 𝑂 → (∃!ℎ ℎ ∈ (𝑂𝐻𝑜) ↔ ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
8	7	rspcv 3577	. . . . 5 ⊢ (𝑂 ∈ 𝐵 → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑜) → ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
9	8	adantl 485	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑜) → ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
10		eusn 4688	. . . . 5 ⊢ (∃!ℎ ℎ ∈ (𝑂𝐻𝑂) ↔ ∃ℎ(𝑂𝐻𝑂) = {ℎ})
11		eqid 2761	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	3	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat)
13		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵)
14	1, 2, 11, 12, 13	catidcl 17697	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → ((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂))
15		fvex 6876	. . . . . . . . . . . . 13 ⊢ ((Id‘𝐶)‘𝑂) ∈ V
16	15	elsn 4596	. . . . . . . . . . . 12 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} ↔ ((Id‘𝐶)‘𝑂) = ℎ)
17		eqcom 2768	. . . . . . . . . . . 12 ⊢ (((Id‘𝐶)‘𝑂) = ℎ ↔ ℎ = ((Id‘𝐶)‘𝑂))
18		sneqbg 4800	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ V → ({ℎ} = {((Id‘𝐶)‘𝑂)} ↔ ℎ = ((Id‘𝐶)‘𝑂)))
19	18	bicomd 225	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ V → (ℎ = ((Id‘𝐶)‘𝑂) ↔ {ℎ} = {((Id‘𝐶)‘𝑂)}))
20	19	elv 3458	. . . . . . . . . . . 12 ⊢ (ℎ = ((Id‘𝐶)‘𝑂) ↔ {ℎ} = {((Id‘𝐶)‘𝑂)})
21	16, 17, 20	3bitri 299	. . . . . . . . . . 11 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} ↔ {ℎ} = {((Id‘𝐶)‘𝑂)})
22	21	biimpi 218	. . . . . . . . . 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 2850	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) ↔ ((Id‘𝐶)‘𝑂) ∈ {ℎ}))
25		eqeq1 2765	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → ((𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)} ↔ {ℎ} = {((Id‘𝐶)‘𝑂)}))
26	23, 24, 25	3imtr4d 296	. . . . . . . 8 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
27	14, 26	syl5 34	. . . . . . 7 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
28	27	exlimiv 1949	. . . . . 6 ⊢ (∃ℎ(𝑂𝐻𝑂) = {ℎ} → (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
29	28	com12 32	. . . . 5 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∃ℎ(𝑂𝐻𝑂) = {ℎ} → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
30	10, 29	biimtrid 244	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∃!ℎ ℎ ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
31	9, 30	syld 47	. . 3 ⊢ (((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑜) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
32	31	expimpd 457	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → ((𝑂 ∈ 𝐵 ∧ ∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑜)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
33	4, 32	mpd 15	1 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∃!weu 2594  ∀wral 3075  Vcvv 3453  {csn 4581  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  Hom chom 17280  Catccat 17679  Idccid 17680  InitOcinito 17997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-cat 17683  df-cid 17684  df-inito 18000

This theorem is referenced by:  2initoinv  18026