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Theorem idafval 18102
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 6906 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 idafval.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4eqtr4di 2795 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6906 . . . . . . . 8 (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶))
7 idafval.1 . . . . . . . 8 1 = (Id‘𝐶)
86, 7eqtr4di 2795 . . . . . . 7 (𝑐 = 𝐶 → (Id‘𝑐) = 1 )
98fveq1d 6908 . . . . . 6 (𝑐 = 𝐶 → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
109oteq3d 4887 . . . . 5 (𝑐 = 𝐶 → ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩ = ⟨𝑥, 𝑥, ( 1𝑥)⟩)
115, 10mpteq12dv 5233 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩) = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
12 df-ida 18100 . . . 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 2789 1 (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cotp 4634  cmpt 5225  cfv 6561  Basecbs 17247  Catccat 17707  Idccid 17708  Idacida 18098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ida 18100
This theorem is referenced by:  idaval  18103  idaf  18108
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