Theorem termoid 17260

Description: For a terminal object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 18-Apr-2020.)

Hypotheses

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

Assertion

Ref	Expression
termoid	⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})

Proof of Theorem termoid

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	istermoi 17258	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂)))
5		oveq1 7157	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜𝐻𝑂) = (𝑂𝐻𝑂))
6	5	eleq2d 2898	. . . . . . 7 ⊢ (𝑜 = 𝑂 → (ℎ ∈ (𝑜𝐻𝑂) ↔ ℎ ∈ (𝑂𝐻𝑂)))
7	6	eubidv 2668	. . . . . 6 ⊢ (𝑜 = 𝑂 → (∃!ℎ ℎ ∈ (𝑜𝐻𝑂) ↔ ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
8	7	rspcv 3618	. . . . 5 ⊢ (𝑂 ∈ 𝐵 → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂) → ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
9	8	adantl 484	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂) → ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
10		eusn 4660	. . . . 5 ⊢ (∃!ℎ ℎ ∈ (𝑂𝐻𝑂) ↔ ∃ℎ(𝑂𝐻𝑂) = {ℎ})
11		eqid 2821	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	3	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat)
13		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵)
14	1, 2, 11, 12, 13	catidcl 16947	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → ((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂))
15		fvex 6678	. . . . . . . . . . . . 13 ⊢ ((Id‘𝐶)‘𝑂) ∈ V
16	15	elsn 4576	. . . . . . . . . . . 12 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} ↔ ((Id‘𝐶)‘𝑂) = ℎ)
17		eqcom 2828	. . . . . . . . . . . 12 ⊢ (((Id‘𝐶)‘𝑂) = ℎ ↔ ℎ = ((Id‘𝐶)‘𝑂))
18		sneqbg 4768	. . . . . . . . . . . . . 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 3500	. . . . . . . . . . . 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 2901	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) ↔ ((Id‘𝐶)‘𝑂) ∈ {ℎ}))
25		eqeq1 2825	. . . . . . . . 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 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
28	27	exlimiv 1927	. . . . . 6 ⊢ (∃ℎ(𝑂𝐻𝑂) = {ℎ} → (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
29	28	com12 32	. . . . 5 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∃ℎ(𝑂𝐻𝑂) = {ℎ} → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
30	10, 29	syl5bi 244	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∃!ℎ ℎ ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
31	9, 30	syld 47	. . 3 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
32	31	expimpd 456	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → ((𝑂 ∈ 𝐵 ∧ ∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
33	4, 32	mpd 15	1 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∃!weu 2649  ∀wral 3138  Vcvv 3495  {csn 4561  ‘cfv 6350  (class class class)co 7150  Basecbs 16477  Hom chom 16570  Catccat 16929  Idccid 16930  TermOctermo 17243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-cat 16933  df-cid 16934  df-termo 17246

This theorem is referenced by:  2termoinv  17271