Theorem termoval 17257

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 17251	. 2 ⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
2		fveq2 6669	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		initoval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	syl6eqr 2874	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fveq2 6669	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6		initoval.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶)
7	5, 6	syl6eqr 2874	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
8	7	oveqd 7172	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑏(Hom ‘𝑐)𝑎) = (𝑏𝐻𝑎))
9	8	eleq2d 2898	. . . . 5 ⊢ (𝑐 = 𝐶 → (ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ℎ ∈ (𝑏𝐻𝑎)))
10	9	eubidv 2668	. . . 4 ⊢ (𝑐 = 𝐶 → (∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)))
11	4, 10	raleqbidv 3401	. . 3 ⊢ (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)))
12	4, 11	rabeqbidv 3485	. 2 ⊢ (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)} = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})
13		initoval.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	3	fvexi 6683	. . . 4 ⊢ 𝐵 ∈ V
15	14	rabex 5234	. . 3 ⊢ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)} ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)} ∈ V)
17	1, 12, 13, 16	fvmptd3 6790	1 ⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ∃!weu 2649  ∀wral 3138  {crab 3142  Vcvv 3494  ‘cfv 6354  (class class class)co 7155  Basecbs 16482  Hom chom 16575  Catccat 16934  TermOctermo 17248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7158  df-termo 17251

This theorem is referenced by:  istermo  17260  istermoi  17263