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Theorem cidfval 17719
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 6921 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6906 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 cidfval.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5eqtr4di 2795 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 fvexd 6921 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8 simpl 482 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
98fveq2d 6910 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10 cidfval.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
119, 10eqtr4di 2795 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12 fvexd 6921 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) ∈ V)
13 simpll 767 . . . . . . . . 9 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑐 = 𝐶)
1413fveq2d 6910 . . . . . . . 8 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
15 cidfval.o . . . . . . . 8 · = (comp‘𝐶)
1614, 15eqtr4di 2795 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = · )
17 simpllr 776 . . . . . . . 8 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → 𝑏 = 𝐵)
18 simplr 769 . . . . . . . . . 10 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → = 𝐻)
1918oveqd 7448 . . . . . . . . 9 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑥𝑥) = (𝑥𝐻𝑥))
2018oveqd 7448 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑦𝑥) = (𝑦𝐻𝑥))
21 simpr 484 . . . . . . . . . . . . . . 15 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → 𝑜 = · )
2221oveqd 7448 . . . . . . . . . . . . . 14 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑦, 𝑥𝑜𝑥) = (⟨𝑦, 𝑥· 𝑥))
2322oveqd 7448 . . . . . . . . . . . . 13 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓))
2423eqeq1d 2739 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → ((𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
2520, 24raleqbidv 3346 . . . . . . . . . . 11 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
2618oveqd 7448 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑥𝑦) = (𝑥𝐻𝑦))
2721oveqd 7448 . . . . . . . . . . . . . 14 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑥, 𝑥𝑜𝑦) = (⟨𝑥, 𝑥· 𝑦))
2827oveqd 7448 . . . . . . . . . . . . 13 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔))
2928eqeq1d 2739 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → ((𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
3026, 29raleqbidv 3346 . . . . . . . . . . 11 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
3125, 30anbi12d 632 . . . . . . . . . 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 3346 . . . . . . . . 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 7391 . . . . . . . 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 5233 . . . . . . 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 3936 . . . . . 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 3936 . . . . 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 3936 . . . 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 17712 . . . 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))))
3937, 38, 5mptfvmpt 7248 . . 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))))
402, 39syl 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))))
411, 40eqtrid 2789 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 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  csb 3899  cop 4632  cmpt 5225  cfv 6561  crio 7387  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-cid 17712
This theorem is referenced by:  cidval  17720  cidfn  17722  catidd  17723  cidpropd  17753
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