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Theorem cidval 16689
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 16688 . 2 (𝜑1 = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))
7 simpr 479 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
87, 7oveq12d 6922 . . 3 ((𝜑𝑥 = 𝑋) → (𝑥𝐻𝑥) = (𝑋𝐻𝑋))
97oveq2d 6920 . . . . . 6 ((𝜑𝑥 = 𝑋) → (𝑦𝐻𝑥) = (𝑦𝐻𝑋))
107opeq2d 4629 . . . . . . . . 9 ((𝜑𝑥 = 𝑋) → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑋⟩)
1110, 7oveq12d 6922 . . . . . . . 8 ((𝜑𝑥 = 𝑋) → (⟨𝑦, 𝑥· 𝑥) = (⟨𝑦, 𝑋· 𝑋))
1211oveqd 6921 . . . . . . 7 ((𝜑𝑥 = 𝑋) → (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = (𝑔(⟨𝑦, 𝑋· 𝑋)𝑓))
1312eqeq1d 2826 . . . . . 6 ((𝜑𝑥 = 𝑋) → ((𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓))
149, 13raleqbidv 3363 . . . . 5 ((𝜑𝑥 = 𝑋) → (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓))
157oveq1d 6919 . . . . . 6 ((𝜑𝑥 = 𝑋) → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
167, 7opeq12d 4630 . . . . . . . . 9 ((𝜑𝑥 = 𝑋) → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
1716oveq1d 6919 . . . . . . . 8 ((𝜑𝑥 = 𝑋) → (⟨𝑥, 𝑥· 𝑦) = (⟨𝑋, 𝑋· 𝑦))
1817oveqd 6921 . . . . . . 7 ((𝜑𝑥 = 𝑋) → (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔))
1918eqeq1d 2826 . . . . . 6 ((𝜑𝑥 = 𝑋) → ((𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
2015, 19raleqbidv 3363 . . . . 5 ((𝜑𝑥 = 𝑋) → (∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
2114, 20anbi12d 626 . . . 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 3194 . . 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 6868 . 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 6869 . . 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 6534 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 386   = wceq 1658  wcel 2166  wral 3116  Vcvv 3413  cop 4402  cfv 6122  crio 6864  (class class class)co 6904  Basecbs 16221  Hom chom 16315  compcco 16316  Catccat 16676  Idccid 16677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-cid 16681
This theorem is referenced by:  catidcl  16694  catlid  16695  catrid  16696
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