Theorem cidfval 17599

Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cidfval.b	⊢ 𝐵 = (Base‘𝐶)
cidfval.h	⊢ 𝐻 = (Hom ‘𝐶)
cidfval.o	⊢ · = (comp‘𝐶)
cidfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
cidfval.i	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
cidfval	⊢ (𝜑 → 1 = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))))

Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝐵   𝐶,𝑓,𝑔,𝑥,𝑦   · ,𝑓,𝑔,𝑥,𝑦   𝑓,𝐻,𝑔,𝑥,𝑦   𝜑,𝑓,𝑔,𝑥,𝑦

Allowed substitution hints:   1 (𝑥,𝑦,𝑓,𝑔)

Proof of Theorem cidfval

Dummy variables 𝑏 𝑐 ℎ 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cidfval.i	. 2 ⊢ 1 = (Id‘𝐶)
2		cidfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fvexd 6849	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4		fveq2 6834	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5		cidfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
6	4, 5	eqtr4di 2789	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7		fvexd 6849	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8		simpl 482	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
9	8	fveq2d 6838	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10		cidfval.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
11	9, 10	eqtr4di 2789	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12		fvexd 6849	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) ∈ V)
13		simpll 766	. . . . . . . . 9 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑐 = 𝐶)
14	13	fveq2d 6838	. . . . . . . 8 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
15		cidfval.o	. . . . . . . 8 ⊢ · = (comp‘𝐶)
16	14, 15	eqtr4di 2789	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = · )
17		simpllr 775	. . . . . . . 8 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → 𝑏 = 𝐵)
18		simplr 768	. . . . . . . . . 10 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ℎ = 𝐻)
19	18	oveqd 7375	. . . . . . . . 9 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑥ℎ𝑥) = (𝑥𝐻𝑥))
20	18	oveqd 7375	. . . . . . . . . . . 12 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑦ℎ𝑥) = (𝑦𝐻𝑥))
21		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → 𝑜 = · )
22	21	oveqd 7375	. . . . . . . . . . . . . 14 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑦, 𝑥⟩𝑜𝑥) = (⟨𝑦, 𝑥⟩ · 𝑥))
23	22	oveqd 7375	. . . . . . . . . . . . 13 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = (𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓))
24	23	eqeq1d 2738	. . . . . . . . . . . 12 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓))
25	20, 24	raleqbidv 3316	. . . . . . . . . . 11 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓))
26	18	oveqd 7375	. . . . . . . . . . . 12 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
27	21	oveqd 7375	. . . . . . . . . . . . . 14 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑥, 𝑥⟩𝑜𝑦) = (⟨𝑥, 𝑥⟩ · 𝑦))
28	27	oveqd 7375	. . . . . . . . . . . . 13 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔))
29	28	eqeq1d 2738	. . . . . . . . . . . 12 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))
30	26, 29	raleqbidv 3316	. . . . . . . . . . 11 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))
31	25, 30	anbi12d 632	. . . . . . . . . 10 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓)))
32	17, 31	raleqbidv 3316	. . . . . . . . 9 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ↔ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓)))
33	19, 32	riotaeqbidv 7318	. . . . . . . 8 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)) = (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓)))
34	17, 33	mpteq12dv 5185	. . . . . . 7 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))) = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))))
35	12, 16, 34	csbied2 3886	. . . . . 6 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))) = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))))
36	7, 11, 35	csbied2 3886	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))) = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))))
37	3, 6, 36	csbied2 3886	. . . 4 ⊢ (𝑐 = 𝐶 → ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))) = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))))
38		df-cid 17592	. . . 4 ⊢ Id = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))))
39	37, 38, 5	mptfvmpt 7174	. . 3 ⊢ (𝐶 ∈ Cat → (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))))
40	2, 39	syl 17	. 2 ⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))))
41	1, 40	eqtrid 2783	1 ⊢ (𝜑 → 1 = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440  ⦋csb 3849  ⟨cop 4586   ↦ cmpt 5179  ‘cfv 6492  ℩crio 7314  (class class class)co 7358  Basecbs 17136  Hom chom 17188  compcco 17189  Catccat 17587  Idccid 17588

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-cid 17592

This theorem is referenced by:  cidval  17600  cidfn  17602  catidd  17603  cidpropd  17633