Theorem cidval 16161

Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-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)
cidval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
cidval	⊢ (𝜑 → ( 1 ‘𝑋) = (℩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,𝑦,𝐵   𝐶,𝑓,𝑔,𝑦   · ,𝑓,𝑔,𝑦   𝑓,𝐻,𝑔,𝑦   𝜑,𝑓,𝑔,𝑦   𝑓,𝑋,𝑔,𝑦

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

Proof of Theorem cidval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cidfval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		cidfval.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		cidfval.o	. . 3 ⊢ · = (comp‘𝐶)
4		cidfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		cidfval.i	. . 3 ⊢ 1 = (Id‘𝐶)
6	1, 2, 3, 4, 5	cidfval 16160	. 2 ⊢ (𝜑 → 1 = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))))
7		simpr 476	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7, 7	oveq12d 6567	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥𝐻𝑥) = (𝑋𝐻𝑋))
9	7	oveq2d 6565	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑦𝐻𝑥) = (𝑦𝐻𝑋))
10	7	opeq2d 4347	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑋⟩)
11	10, 7	oveq12d 6567	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨𝑦, 𝑥⟩ · 𝑥) = (⟨𝑦, 𝑋⟩ · 𝑋))
12	11	oveqd 6566	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = (𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓))
13	12	eqeq1d 2612	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
14	9, 13	raleqbidv 3129	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
15	7	oveq1d 6564	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16	7, 7	opeq12d 4348	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
17	16	oveq1d 6564	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨𝑥, 𝑥⟩ · 𝑦) = (⟨𝑋, 𝑋⟩ · 𝑦))
18	17	oveqd 6566	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔))
19	18	eqeq1d 2612	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
20	15, 19	raleqbidv 3129	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
21	14, 20	anbi12d 743	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)))
22	21	ralbidv 2969	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓) ↔ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)))
23	8, 22	riotaeqbidv 6514	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓)) = (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)))
24		cidval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
25		riotaex 6515	. . 3 ⊢ (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)) ∈ V
26	25	a1i 11	. 2 ⊢ (𝜑 → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)) ∈ V)
27	6, 23, 24, 26	fvmptd 6197	1 ⊢ (𝜑 → ( 1 ‘𝑋) = (℩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 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149

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-rep 4699  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-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  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-iun 4457  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-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-cid 16153

This theorem is referenced by:  catidcl  16166  catlid  16167  catrid  16168