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Theorem cidfval 16602
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 6390 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6375 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 cidfval.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5syl6eqr 2817 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 fvexd 6390 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8 simpl 474 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
98fveq2d 6379 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10 cidfval.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
119, 10syl6eqr 2817 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12 fvexd 6390 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) ∈ V)
13 simpll 783 . . . . . . . . 9 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑐 = 𝐶)
1413fveq2d 6379 . . . . . . . 8 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
15 cidfval.o . . . . . . . 8 · = (comp‘𝐶)
1614, 15syl6eqr 2817 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = · )
17 simpllr 793 . . . . . . . 8 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → 𝑏 = 𝐵)
18 simplr 785 . . . . . . . . . 10 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → = 𝐻)
1918oveqd 6859 . . . . . . . . 9 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑥𝑥) = (𝑥𝐻𝑥))
2018oveqd 6859 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑦𝑥) = (𝑦𝐻𝑥))
21 simpr 477 . . . . . . . . . . . . . . 15 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → 𝑜 = · )
2221oveqd 6859 . . . . . . . . . . . . . 14 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑦, 𝑥𝑜𝑥) = (⟨𝑦, 𝑥· 𝑥))
2322oveqd 6859 . . . . . . . . . . . . 13 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓))
2423eqeq1d 2767 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → ((𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
2520, 24raleqbidv 3300 . . . . . . . . . . 11 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
2618oveqd 6859 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑥𝑦) = (𝑥𝐻𝑦))
2721oveqd 6859 . . . . . . . . . . . . . 14 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑥, 𝑥𝑜𝑦) = (⟨𝑥, 𝑥· 𝑦))
2827oveqd 6859 . . . . . . . . . . . . 13 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔))
2928eqeq1d 2767 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → ((𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
3026, 29raleqbidv 3300 . . . . . . . . . . 11 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
3125, 30anbi12d 624 . . . . . . . . . 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 3300 . . . . . . . . 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 6806 . . . . . . . 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 4892 . . . . . . 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 3719 . . . . . 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 3719 . . . . 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 3719 . . . 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 16595 . . . 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 6683 . . 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, 40syl5eq 2811 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 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  csb 3691  cop 4340  cmpt 4888  cfv 6068  crio 6802  (class class class)co 6842  Basecbs 16130  Hom chom 16225  compcco 16226  Catccat 16590  Idccid 16591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-cid 16595
This theorem is referenced by:  cidval  16603  cidfn  16605  catidd  16606  cidpropd  16635
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