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Theorem cidfval 16939
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 6678 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6663 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 cidfval.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5syl6eqr 2872 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 fvexd 6678 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8 simpl 485 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
98fveq2d 6667 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10 cidfval.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
119, 10syl6eqr 2872 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12 fvexd 6678 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) ∈ V)
13 simpll 765 . . . . . . . . 9 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑐 = 𝐶)
1413fveq2d 6667 . . . . . . . 8 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
15 cidfval.o . . . . . . . 8 · = (comp‘𝐶)
1614, 15syl6eqr 2872 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = · )
17 simpllr 774 . . . . . . . 8 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → 𝑏 = 𝐵)
18 simplr 767 . . . . . . . . . 10 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → = 𝐻)
1918oveqd 7165 . . . . . . . . 9 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑥𝑥) = (𝑥𝐻𝑥))
2018oveqd 7165 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑦𝑥) = (𝑦𝐻𝑥))
21 simpr 487 . . . . . . . . . . . . . . 15 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → 𝑜 = · )
2221oveqd 7165 . . . . . . . . . . . . . 14 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑦, 𝑥𝑜𝑥) = (⟨𝑦, 𝑥· 𝑥))
2322oveqd 7165 . . . . . . . . . . . . 13 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓))
2423eqeq1d 2821 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → ((𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
2520, 24raleqbidv 3400 . . . . . . . . . . 11 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
2618oveqd 7165 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑥𝑦) = (𝑥𝐻𝑦))
2721oveqd 7165 . . . . . . . . . . . . . 14 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑥, 𝑥𝑜𝑦) = (⟨𝑥, 𝑥· 𝑦))
2827oveqd 7165 . . . . . . . . . . . . 13 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔))
2928eqeq1d 2821 . . . . . . . . . . . 12 ((((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) ∧ 𝑜 = · ) → ((𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
3026, 29raleqbidv 3400 . . . . . . . . . . 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 3400 . . . . . . . . 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 7109 . . . . . . . 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 5142 . . . . . . 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 3918 . . . . . 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 3918 . . . . 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 3918 . . . 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 16932 . . . 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 6982 . . 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 2866 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 398   = wceq 1530  wcel 2107  wral 3136  Vcvv 3493  csb 3881  cop 4565  cmpt 5137  cfv 6348  crio 7105  (class class class)co 7148  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Idccid 16928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-cid 16932
This theorem is referenced by:  cidval  16940  cidfn  16942  catidd  16943  cidpropd  16972
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