Theorem istermoi 17969

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 17963	. . . . 5 ⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})
5	4	eleq2d 2815	. . . 4 ⊢ (𝜑 → (𝑂 ∈ (TermO‘𝐶) ↔ 𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)}))
6		elrabi 3657	. . . 4 ⊢ (𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)} → 𝑂 ∈ 𝐵)
7	5, 6	biimtrdi 253	. . 3 ⊢ (𝜑 → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵))
8	7	imp 406	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ 𝐵)
9	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat)
10		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵)
11	2, 3, 9, 10	istermo 17966	. . . 4 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (TermO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))
12	11	biimpd 229	. . 3 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (TermO‘𝐶) → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))
13	12	impancom 451	. 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))
14	8, 13	jcai 516	1 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃!weu 2562  ∀wral 3045  {crab 3408  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Hom chom 17238  Catccat 17632  TermOctermo 17951

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-termo 17954

This theorem is referenced by:  termoid  17971  termoo  17977  termoeu1  17987  termoo2  49226