Theorem cidval 17765

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 17764	. 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 490	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7, 7	oveq12d 7431	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥𝐻𝑥) = (𝑋𝐻𝑋))
9	7	oveq2d 7429	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑦𝐻𝑥) = (𝑦𝐻𝑋))
10	7	opeq2d 4840	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑋⟩)
11	10, 7	oveq12d 7431	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨𝑦, 𝑥⟩ · 𝑥) = (⟨𝑦, 𝑋⟩ · 𝑋))
12	11	oveqd 7430	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = (𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓))
13	12	eqeq1d 2762	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
14	9, 13	raleqbidv 3334	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
15	7	oveq1d 7428	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16	7, 7	opeq12d 4841	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
17	16	oveq1d 7428	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨𝑥, 𝑥⟩ · 𝑦) = (⟨𝑋, 𝑋⟩ · 𝑦))
18	17	oveqd 7430	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔))
19	18	eqeq1d 2762	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
20	15, 19	raleqbidv 3334	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
21	14, 20	anbi12d 644	. . . 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 3185	. . 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 7373	. 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 7374	. . 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 6994	1 ⊢ (𝜑 → ( 1 ‘𝑋) = (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450  ⟨cop 4590  ‘cfv 6533  ℩crio 7369  (class class class)co 7413  Basecbs 17301  Hom chom 17353  compcco 17354  Catccat 17752  Idccid 17753

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-cid 17757

This theorem is used by:  catidcl  17770  catlid  17771  catrid  17772