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Theorem cidval 16332
 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 (𝜑𝐶 ∈ Cat)
cidfval.i 1 = (Id‘𝐶)
cidval.x (𝜑𝑋𝐵)
Assertion
Ref Expression
cidval (𝜑 → ( 1𝑋) = (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀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.
StepHypRef 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 (𝜑𝐶 ∈ Cat)
5 cidfval.i . . 3 1 = (Id‘𝐶)
61, 2, 3, 4, 5cidfval 16331 . 2 (𝜑1 = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))
7 simpr 477 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
87, 7oveq12d 6665 . . 3 ((𝜑𝑥 = 𝑋) → (𝑥𝐻𝑥) = (𝑋𝐻𝑋))
97oveq2d 6663 . . . . . 6 ((𝜑𝑥 = 𝑋) → (𝑦𝐻𝑥) = (𝑦𝐻𝑋))
107opeq2d 4407 . . . . . . . . 9 ((𝜑𝑥 = 𝑋) → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑋⟩)
1110, 7oveq12d 6665 . . . . . . . 8 ((𝜑𝑥 = 𝑋) → (⟨𝑦, 𝑥· 𝑥) = (⟨𝑦, 𝑋· 𝑋))
1211oveqd 6664 . . . . . . 7 ((𝜑𝑥 = 𝑋) → (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = (𝑔(⟨𝑦, 𝑋· 𝑋)𝑓))
1312eqeq1d 2623 . . . . . 6 ((𝜑𝑥 = 𝑋) → ((𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓))
149, 13raleqbidv 3150 . . . . 5 ((𝜑𝑥 = 𝑋) → (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓))
157oveq1d 6662 . . . . . 6 ((𝜑𝑥 = 𝑋) → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
167, 7opeq12d 4408 . . . . . . . . 9 ((𝜑𝑥 = 𝑋) → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
1716oveq1d 6662 . . . . . . . 8 ((𝜑𝑥 = 𝑋) → (⟨𝑥, 𝑥· 𝑦) = (⟨𝑋, 𝑋· 𝑦))
1817oveqd 6664 . . . . . . 7 ((𝜑𝑥 = 𝑋) → (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔))
1918eqeq1d 2623 . . . . . 6 ((𝜑𝑥 = 𝑋) → ((𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
2015, 19raleqbidv 3150 . . . . 5 ((𝜑𝑥 = 𝑋) → (∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
2114, 20anbi12d 747 . . . 4 ((𝜑𝑥 = 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)))
2221ralbidv 2985 . . 3 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ↔ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)))
238, 22riotaeqbidv 6611 . 2 ((𝜑𝑥 = 𝑋) → (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓)) = (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)))
24 cidval.x . 2 (𝜑𝑋𝐵)
25 riotaex 6612 . . 3 (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)) ∈ V
2625a1i 11 . 2 (𝜑 → (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)) ∈ V)
276, 23, 24, 26fvmptd 6286 1 (𝜑 → ( 1𝑋) = (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 1482   ∈ wcel 1989  ∀wral 2911  Vcvv 3198  ⟨cop 4181  ‘cfv 5886  ℩crio 6607  (class class class)co 6647  Basecbs 15851  Hom chom 15946  compcco 15947  Catccat 16319  Idccid 16320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-cid 16324 This theorem is referenced by:  catidcl  16337  catlid  16338  catrid  16339
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