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Theorem idafval 18111
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 6907 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 idafval.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4eqtr4di 2793 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6907 . . . . . . . 8 (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶))
7 idafval.1 . . . . . . . 8 1 = (Id‘𝐶)
86, 7eqtr4di 2793 . . . . . . 7 (𝑐 = 𝐶 → (Id‘𝑐) = 1 )
98fveq1d 6909 . . . . . 6 (𝑐 = 𝐶 → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
109oteq3d 4892 . . . . 5 (𝑐 = 𝐶 → ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩ = ⟨𝑥, 𝑥, ( 1𝑥)⟩)
115, 10mpteq12dv 5239 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩) = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
12 df-ida 18109 . . . 4 Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
1311, 12, 4mptfvmpt 7248 . . 3 (𝐶 ∈ Cat → (Ida𝐶) = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
142, 13syl 17 . 2 (𝜑 → (Ida𝐶) = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
151, 14eqtrid 2787 1 (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cotp 4639  cmpt 5231  cfv 6563  Basecbs 17245  Catccat 17709  Idccid 17710  Idacida 18107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ida 18109
This theorem is referenced by:  idaval  18112  idaf  18117
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