Theorem istermo 17904

Description: The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020.)

Hypotheses

Ref	Expression
isinito.b	⊢ 𝐵 = (Base‘𝐶)
isinito.h	⊢ 𝐻 = (Hom ‘𝐶)
isinito.c	⊢ (𝜑 → 𝐶 ∈ Cat)
isinito.i	⊢ (𝜑 → 𝐼 ∈ 𝐵)

Assertion

Ref	Expression
istermo	⊢ (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼)))

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

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

Proof of Theorem istermo

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isinito.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		isinito.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		isinito.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
4	1, 2, 3	termoval 17901	. . 3 ⊢ (𝜑 → (TermO‘𝐶) = {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖)})
5	4	eleq2d 2817	. 2 ⊢ (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖)}))
6		isinito.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
7		oveq2 7354	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑏𝐻𝑖) = (𝑏𝐻𝐼))
8	7	eleq2d 2817	. . . . . 6 ⊢ (𝑖 = 𝐼 → (ℎ ∈ (𝑏𝐻𝑖) ↔ ℎ ∈ (𝑏𝐻𝐼)))
9	8	eubidv 2581	. . . . 5 ⊢ (𝑖 = 𝐼 → (∃!ℎ ℎ ∈ (𝑏𝐻𝑖) ↔ ∃!ℎ ℎ ∈ (𝑏𝐻𝐼)))
10	9	ralbidv 3155	. . . 4 ⊢ (𝑖 = 𝐼 → (∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼)))
11	10	elrab3 3643	. . 3 ⊢ (𝐼 ∈ 𝐵 → (𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼)))
12	6, 11	syl 17	. 2 ⊢ (𝜑 → (𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼)))
13	5, 12	bitrd 279	1 ⊢ (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∃!weu 2563  ∀wral 3047  {crab 3395  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  Hom chom 17172  Catccat 17570  TermOctermo 17889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-termo 17892

This theorem is referenced by:  istermoi  17907  zrtermorngc  20558  zrtermoringc  20590  termcterm  49613  termc2  49618