Theorem cidval 17638

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 17637	. 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7, 7	oveq12d 7405	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥𝐻𝑥) = (𝑋𝐻𝑋))
9	7	oveq2d 7403	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑦𝐻𝑥) = (𝑦𝐻𝑋))
10	7	opeq2d 4844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑋⟩)
11	10, 7	oveq12d 7405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨𝑦, 𝑥⟩ · 𝑥) = (⟨𝑦, 𝑋⟩ · 𝑋))
12	11	oveqd 7404	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = (𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓))
13	12	eqeq1d 2731	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
14	9, 13	raleqbidv 3319	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
15	7	oveq1d 7402	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16	7, 7	opeq12d 4845	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
17	16	oveq1d 7402	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨𝑥, 𝑥⟩ · 𝑦) = (⟨𝑋, 𝑋⟩ · 𝑦))
18	17	oveqd 7404	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔))
19	18	eqeq1d 2731	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
20	15, 19	raleqbidv 3319	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
21	14, 20	anbi12d 632	. . . 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 3156	. . 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 7347	. 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 7348	. . 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 6975	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 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447  ⟨cop 4595  ‘cfv 6511  ℩crio 7343  (class class class)co 7387  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17625  Idccid 17626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-cid 17630

This theorem is referenced by:  catidcl  17643  catlid  17644  catrid  17645