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Theorem cidfval 16530
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 (𝜑𝐶 ∈ Cat)
cidfval.i 1 = (Id‘𝐶)
Assertion
Ref Expression
cidfval (𝜑1 = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀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 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cidfval.i . 2 1 = (Id‘𝐶)
2 cidfval.c . . 3 (𝜑𝐶 ∈ Cat)
3 fvexd 6356 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6344 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 cidfval.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5syl6eqr 2804 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 fvexd 6356 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8 simpl 474 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
98fveq2d 6348 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10 cidfval.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
119, 10syl6eqr 2804 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12 fvexd 6356 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) ∈ V)
13 simpll 807 . . . . . . . . 9 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑐 = 𝐶)
1413fveq2d 6348 . . . . . . . 8 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
15 cidfval.o . . . . . . . 8 · = (comp‘𝐶)
1614, 15syl6eqr 2804 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = · )
17 simpllr 817 . . . . . . . 8 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → 𝑏 = 𝐵)
18 simplr 809 . . . . . . . . . 10 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → = 𝐻)
1918oveqd 6822 . . . . . . . . 9 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑥𝑥) = (𝑥𝐻𝑥))
2018oveqd 6822 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑦𝑥) = (𝑦𝐻𝑥))
21 simpr 479 . . . . . . . . . . . . . . 15 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → 𝑜 = · )
2221oveqd 6822 . . . . . . . . . . . . . 14 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑦, 𝑥𝑜𝑥) = (⟨𝑦, 𝑥· 𝑥))
2322oveqd 6822 . . . . . . . . . . . . 13 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓))
2423eqeq1d 2754 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → ((𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
2520, 24raleqbidv 3283 . . . . . . . . . . 11 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
2618oveqd 6822 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑥𝑦) = (𝑥𝐻𝑦))
2721oveqd 6822 . . . . . . . . . . . . . 14 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑥, 𝑥𝑜𝑦) = (⟨𝑥, 𝑥· 𝑦))
2827oveqd 6822 . . . . . . . . . . . . 13 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔))
2928eqeq1d 2754 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → ((𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
3026, 29raleqbidv 3283 . . . . . . . . . . 11 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
3125, 30anbi12d 749 . . . . . . . . . 10 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → ((∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓)))
3217, 31raleqbidv 3283 . . . . . . . . 9 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓) ↔ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓)))
3319, 32riotaeqbidv 6769 . . . . . . . 8 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)) = (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓)))
3417, 33mpteq12dv 4877 . . . . . . 7 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))) = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))
3512, 16, 34csbied2 3694 . . . . . 6 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))) = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))
367, 11, 35csbied2 3694 . . . . 5 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))) = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))
373, 6, 36csbied2 3694 . . . 4 (𝑐 = 𝐶(Base‘𝑐) / 𝑏(Hom ‘𝑐) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))) = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))
38 df-cid 16523 . . . 4 Id = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏(Hom ‘𝑐) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))))
39 fvex 6354 . . . . . 6 (Base‘𝐶) ∈ V
405, 39eqeltri 2827 . . . . 5 𝐵 ∈ V
4140mptex 6642 . . . 4 (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))) ∈ V
4237, 38, 41fvmpt 6436 . . 3 (𝐶 ∈ Cat → (Id‘𝐶) = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))
432, 42syl 17 . 2 (𝜑 → (Id‘𝐶) = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))
441, 43syl5eq 2798 1 (𝜑1 = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  csb 3666  cop 4319  cmpt 4873  cfv 6041  crio 6765  (class class class)co 6805  Basecbs 16051  Hom chom 16146  compcco 16147  Catccat 16518  Idccid 16519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-cid 16523
This theorem is referenced by:  cidval  16531  cidfn  16533  catidd  16534  cidpropd  16563
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