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Theorem idaval 18047
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)
idaval.x (𝜑𝑋𝐵)
Assertion
Ref Expression
idaval (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)

Proof of Theorem idaval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 idafval.i . . 3 𝐼 = (Ida𝐶)
2 idafval.b . . 3 𝐵 = (Base‘𝐶)
3 idafval.c . . 3 (𝜑𝐶 ∈ Cat)
4 idafval.1 . . 3 1 = (Id‘𝐶)
51, 2, 3, 4idafval 18046 . 2 (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
6 simpr 484 . . 3 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
76fveq2d 6901 . . 3 ((𝜑𝑥 = 𝑋) → ( 1𝑥) = ( 1𝑋))
86, 6, 7oteq123d 4889 . 2 ((𝜑𝑥 = 𝑋) → ⟨𝑥, 𝑥, ( 1𝑥)⟩ = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
9 idaval.x . 2 (𝜑𝑋𝐵)
10 otex 5467 . . 3 𝑋, 𝑋, ( 1𝑋)⟩ ∈ V
1110a1i 11 . 2 (𝜑 → ⟨𝑋, 𝑋, ( 1𝑋)⟩ ∈ V)
125, 8, 9, 11fvmptd 7012 1 (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  cotp 4637  cfv 6548  Basecbs 17180  Catccat 17644  Idccid 17645  Idacida 18042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ida 18044
This theorem is referenced by:  ida2  18048  idahom  18049
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