Theorem termoid 17927

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 17925	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂)))
5		oveq1 7360	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜𝐻𝑂) = (𝑂𝐻𝑂))
6	5	eleq2d 2814	. . . . . . 7 ⊢ (𝑜 = 𝑂 → (ℎ ∈ (𝑜𝐻𝑂) ↔ ℎ ∈ (𝑂𝐻𝑂)))
7	6	eubidv 2579	. . . . . 6 ⊢ (𝑜 = 𝑂 → (∃!ℎ ℎ ∈ (𝑜𝐻𝑂) ↔ ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
8	7	rspcv 3575	. . . . 5 ⊢ (𝑂 ∈ 𝐵 → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂) → ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
9	8	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂) → ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
10		eusn 4684	. . . . 5 ⊢ (∃!ℎ ℎ ∈ (𝑂𝐻𝑂) ↔ ∃ℎ(𝑂𝐻𝑂) = {ℎ})
11		eqid 2729	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	3	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat)
13		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵)
14	1, 2, 11, 12, 13	catidcl 17606	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → ((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂))
15		fvex 6839	. . . . . . . . . . . . 13 ⊢ ((Id‘𝐶)‘𝑂) ∈ V
16	15	elsn 4594	. . . . . . . . . . . 12 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} ↔ ((Id‘𝐶)‘𝑂) = ℎ)
17		eqcom 2736	. . . . . . . . . . . 12 ⊢ (((Id‘𝐶)‘𝑂) = ℎ ↔ ℎ = ((Id‘𝐶)‘𝑂))
18		sneqbg 4797	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ V → ({ℎ} = {((Id‘𝐶)‘𝑂)} ↔ ℎ = ((Id‘𝐶)‘𝑂)))
19	18	bicomd 223	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ V → (ℎ = ((Id‘𝐶)‘𝑂) ↔ {ℎ} = {((Id‘𝐶)‘𝑂)}))
20	19	elv 3443	. . . . . . . . . . . 12 ⊢ (ℎ = ((Id‘𝐶)‘𝑂) ↔ {ℎ} = {((Id‘𝐶)‘𝑂)})
21	16, 17, 20	3bitri 297	. . . . . . . . . . 11 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} ↔ {ℎ} = {((Id‘𝐶)‘𝑂)})
22	21	biimpi 216	. . . . . . . . . 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 2817	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) ↔ ((Id‘𝐶)‘𝑂) ∈ {ℎ}))
25		eqeq1 2733	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → ((𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)} ↔ {ℎ} = {((Id‘𝐶)‘𝑂)}))
26	23, 24, 25	3imtr4d 294	. . . . . . . 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 1930	. . . . . 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	biimtrid 242	. . . 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 453	. 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 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃!weu 2561  ∀wral 3044  Vcvv 3438  {csn 4579  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  Hom chom 17190  Catccat 17588  Idccid 17589  TermOctermo 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-rep 5221  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-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  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-iun 4946  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-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-cat 17592  df-cid 17593  df-termo 17910

This theorem is referenced by:  2termoinv  17942