Theorem termoval 16471

Description: The value of the terminal object function, i.e. the set of all terminal objects of a category. (Contributed by AV, 3-Apr-2020.)

Hypotheses

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

Assertion

Ref	Expression
termoval	⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})

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

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

Proof of Theorem termoval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-termo 16465	. . 3 ⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
2	1	a1i 11	. 2 ⊢ (𝜑 → TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)}))
3		fveq2 6103	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		initoval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	syl6eqr 2662	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6		fveq2 6103	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7		initoval.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐶)
8	6, 7	syl6eqr 2662	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
9	8	oveqd 6566	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑏(Hom ‘𝑐)𝑎) = (𝑏𝐻𝑎))
10	9	eleq2d 2673	. . . . . 6 ⊢ (𝑐 = 𝐶 → (ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ℎ ∈ (𝑏𝐻𝑎)))
11	10	eubidv 2478	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)))
12	5, 11	raleqbidv 3129	. . . 4 ⊢ (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)))
13	5, 12	rabeqbidv 3168	. . 3 ⊢ (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)} = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})
14	13	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)} = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})
15		initoval.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
16		fvex 6113	. . . . 5 ⊢ (Base‘𝐶) ∈ V
17	4, 16	eqeltri 2684	. . . 4 ⊢ 𝐵 ∈ V
18	17	rabex 4740	. . 3 ⊢ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)} ∈ V
19	18	a1i 11	. 2 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)} ∈ V)
20	2, 14, 15, 19	fvmptd 6197	1 ⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  ∀wral 2896  {crab 2900  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  Catccat 16148  TermOctermo 16462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-termo 16465

This theorem is referenced by:  istermo  16474  istermoi  16477