Theorem termoid 16656

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 16654	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂)))
5		oveq1 6657	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜𝐻𝑂) = (𝑂𝐻𝑂))
6	5	eleq2d 2687	. . . . . . 7 ⊢ (𝑜 = 𝑂 → (ℎ ∈ (𝑜𝐻𝑂) ↔ ℎ ∈ (𝑂𝐻𝑂)))
7	6	eubidv 2490	. . . . . 6 ⊢ (𝑜 = 𝑂 → (∃!ℎ ℎ ∈ (𝑜𝐻𝑂) ↔ ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
8	7	rspcv 3305	. . . . 5 ⊢ (𝑂 ∈ 𝐵 → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂) → ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
9	8	adantl 482	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂) → ∃!ℎ ℎ ∈ (𝑂𝐻𝑂)))
10		eusn 4265	. . . . 5 ⊢ (∃!ℎ ℎ ∈ (𝑂𝐻𝑂) ↔ ∃ℎ(𝑂𝐻𝑂) = {ℎ})
11		eqid 2622	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	3	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat)
13		simpr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵)
14	1, 2, 11, 12, 13	catidcl 16343	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → ((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂))
15		fvex 6201	. . . . . . . . . . . . 13 ⊢ ((Id‘𝐶)‘𝑂) ∈ V
16	15	elsn 4192	. . . . . . . . . . . 12 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} ↔ ((Id‘𝐶)‘𝑂) = ℎ)
17		eqcom 2629	. . . . . . . . . . . 12 ⊢ (((Id‘𝐶)‘𝑂) = ℎ ↔ ℎ = ((Id‘𝐶)‘𝑂))
18		vex 3203	. . . . . . . . . . . . 13 ⊢ ℎ ∈ V
19		sneqbg 4374	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ V → ({ℎ} = {((Id‘𝐶)‘𝑂)} ↔ ℎ = ((Id‘𝐶)‘𝑂)))
20	19	bicomd 213	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ V → (ℎ = ((Id‘𝐶)‘𝑂) ↔ {ℎ} = {((Id‘𝐶)‘𝑂)}))
21	18, 20	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℎ = ((Id‘𝐶)‘𝑂) ↔ {ℎ} = {((Id‘𝐶)‘𝑂)})
22	16, 17, 21	3bitri 286	. . . . . . . . . . 11 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} ↔ {ℎ} = {((Id‘𝐶)‘𝑂)})
23	22	biimpi 206	. . . . . . . . . 10 ⊢ (((Id‘𝐶)‘𝑂) ∈ {ℎ} → {ℎ} = {((Id‘𝐶)‘𝑂)})
24	23	a1i 11	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((Id‘𝐶)‘𝑂) ∈ {ℎ} → {ℎ} = {((Id‘𝐶)‘𝑂)}))
25		eleq2 2690	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) ↔ ((Id‘𝐶)‘𝑂) ∈ {ℎ}))
26		eqeq1 2626	. . . . . . . . 9 ⊢ ((𝑂𝐻𝑂) = {ℎ} → ((𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)} ↔ {ℎ} = {((Id‘𝐶)‘𝑂)}))
27	24, 25, 26	3imtr4d 283	. . . . . . . 8 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
28	14, 27	syl5 34	. . . . . . 7 ⊢ ((𝑂𝐻𝑂) = {ℎ} → (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
29	28	exlimiv 1858	. . . . . 6 ⊢ (∃ℎ(𝑂𝐻𝑂) = {ℎ} → (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
30	29	com12 32	. . . . 5 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∃ℎ(𝑂𝐻𝑂) = {ℎ} → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
31	10, 30	syl5bi 232	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∃!ℎ ℎ ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
32	9, 31	syld 47	. . 3 ⊢ (((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂 ∈ 𝐵) → (∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
33	32	expimpd 629	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → ((𝑂 ∈ 𝐵 ∧ ∀𝑜 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑜𝐻𝑂)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
34	4, 33	mpd 15	1 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  ∀wral 2912  Vcvv 3200  {csn 4177  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Hom chom 15952  Catccat 16325  Idccid 16326  TermOctermo 16639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-cat 16329  df-cid 16330  df-termo 16642

This theorem is referenced by:  2termoinv  16667