Theorem istermoi 18083

Description: Implication of a class being a terminal object. (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
istermoi	⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))

Distinct variable groups:   𝐵,𝑏   𝐶,𝑏,ℎ   𝑂,𝑏,ℎ

Allowed substitution hints:   𝜑(ℎ, 𝑏)   𝐵(ℎ)   𝐻(ℎ, 𝑏)

Proof of Theorem istermoi

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isinitoi.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		isinitoi.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
3		isinitoi.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
4	1, 2, 3	termoval 18077	. . . . 5 ⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})
5	4	eleq2d 2851	. . . 4 ⊢ (𝜑 → (𝑂 ∈ (TermO‘𝐶) ↔ 𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)}))
6		elrabi 3648	. . . 4 ⊢ (𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)} → 𝑂 ∈ 𝐵)
7	5, 6	biimtrdi 256	. . 3 ⊢ (𝜑 → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵))
8	7	imp 412	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ 𝐵)
9	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat)
10		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵)
11	2, 3, 9, 10	istermo 18080	. . . 4 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (TermO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))
12	11	biimpd 232	. . 3 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (TermO‘𝐶) → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))
13	12	impancom 457	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))
14	8, 13	jcai 526	1 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃!weu 2598  ∀wral 3081  {crab 3418  ‘cfv 6540  (class class class)co 7419  Basecbs 17295  Hom chom 17347  Catccat 17746  TermOctermo 18065

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-termo 18068

This theorem is used by:  termoid  18085  termoo  18091  termoeu1  18101  termoo2  50087