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Theorem idafval 16530
 Description: Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
idafval.i 𝐼 = (Ida𝐶)
idafval.b 𝐵 = (Base‘𝐶)
idafval.c (𝜑𝐶 ∈ Cat)
idafval.1 1 = (Id‘𝐶)
Assertion
Ref Expression
idafval (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
Distinct variable groups:   𝑥, 1   𝑥,𝐵   𝑥,𝐶   𝑥,𝐼   𝜑,𝑥

Proof of Theorem idafval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 idafval.i . 2 𝐼 = (Ida𝐶)
2 idafval.c . . 3 (𝜑𝐶 ∈ Cat)
3 fveq2 6103 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 idafval.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2662 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6103 . . . . . . . 8 (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶))
7 idafval.1 . . . . . . . 8 1 = (Id‘𝐶)
86, 7syl6eqr 2662 . . . . . . 7 (𝑐 = 𝐶 → (Id‘𝑐) = 1 )
98fveq1d 6105 . . . . . 6 (𝑐 = 𝐶 → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
109oteq3d 4354 . . . . 5 (𝑐 = 𝐶 → ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩ = ⟨𝑥, 𝑥, ( 1𝑥)⟩)
115, 10mpteq12dv 4663 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩) = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
12 df-ida 16528 . . . 4 Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
13 fvex 6113 . . . . . 6 (Base‘𝐶) ∈ V
144, 13eqeltri 2684 . . . . 5 𝐵 ∈ V
1514mptex 6390 . . . 4 (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩) ∈ V
1611, 12, 15fvmpt 6191 . . 3 (𝐶 ∈ Cat → (Ida𝐶) = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
172, 16syl 17 . 2 (𝜑 → (Ida𝐶) = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
181, 17syl5eq 2656 1 (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cotp 4133   ↦ cmpt 4643  ‘cfv 5804  Basecbs 15695  Catccat 16148  Idccid 16149  Idacida 16526 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ida 16528 This theorem is referenced by:  idaval  16531  idaf  16536
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