Theorem istermoi 17715

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 17709	. . . . 5 ⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})
5	4	eleq2d 2824	. . . 4 ⊢ (𝜑 → (𝑂 ∈ (TermO‘𝐶) ↔ 𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)}))
6		elrabi 3618	. . . 4 ⊢ (𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)} → 𝑂 ∈ 𝐵)
7	5, 6	syl6bi 252	. . 3 ⊢ (𝜑 → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵))
8	7	imp 407	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ 𝐵)
9	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat)
10		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵)
11	2, 3, 9, 10	istermo 17712	. . . 4 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (TermO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))
12	11	biimpd 228	. . 3 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (TermO‘𝐶) → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))
13	12	impancom 452	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))
14	8, 13	jcai 517	1 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃!weu 2568  ∀wral 3064  {crab 3068  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  Catccat 17373  TermOctermo 17697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-termo 17700

This theorem is referenced by:  termoid  17717  termoo  17723  termoeu1  17733