Theorem cidval 17641

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 17640	. 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7, 7	oveq12d 7381	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥𝐻𝑥) = (𝑋𝐻𝑋))
9	7	oveq2d 7379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑦𝐻𝑥) = (𝑦𝐻𝑋))
10	7	opeq2d 4818	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑋⟩)
11	10, 7	oveq12d 7381	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨𝑦, 𝑥⟩ · 𝑥) = (⟨𝑦, 𝑋⟩ · 𝑋))
12	11	oveqd 7380	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = (𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓))
13	12	eqeq1d 2742	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
14	9, 13	raleqbidv 3314	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
15	7	oveq1d 7378	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16	7, 7	opeq12d 4819	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
17	16	oveq1d 7378	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨𝑥, 𝑥⟩ · 𝑦) = (⟨𝑋, 𝑋⟩ · 𝑦))
18	17	oveqd 7380	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔))
19	18	eqeq1d 2742	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
20	15, 19	raleqbidv 3314	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
21	14, 20	anbi12d 638	. . . 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 3163	. . 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 7323	. 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 7324	. . 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 6950	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 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  Vcvv 3432  ⟨cop 4568  ‘cfv 6492  ℩crio 7319  (class class class)co 7363  Basecbs 17177  Hom chom 17229  compcco 17230  Catccat 17628  Idccid 17629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7320  df-ov 7366  df-cid 17633

This theorem is referenced by:  catidcl  17646  catlid  17647  catrid  17648