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Theorem idfu1 17931
Description: Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
idfuval.i 𝐼 = (idfunc𝐶)
idfuval.b 𝐵 = (Base‘𝐶)
idfuval.c (𝜑𝐶 ∈ Cat)
idfu1.x (𝜑𝑋𝐵)
Assertion
Ref Expression
idfu1 (𝜑 → ((1st𝐼)‘𝑋) = 𝑋)

Proof of Theorem idfu1
StepHypRef Expression
1 idfuval.i . . . 4 𝐼 = (idfunc𝐶)
2 idfuval.b . . . 4 𝐵 = (Base‘𝐶)
3 idfuval.c . . . 4 (𝜑𝐶 ∈ Cat)
41, 2, 3idfu1st 17930 . . 3 (𝜑 → (1st𝐼) = ( I ↾ 𝐵))
54fveq1d 6909 . 2 (𝜑 → ((1st𝐼)‘𝑋) = (( I ↾ 𝐵)‘𝑋))
6 idfu1.x . . 3 (𝜑𝑋𝐵)
7 fvresi 7193 . . 3 (𝑋𝐵 → (( I ↾ 𝐵)‘𝑋) = 𝑋)
86, 7syl 17 . 2 (𝜑 → (( I ↾ 𝐵)‘𝑋) = 𝑋)
95, 8eqtrd 2775 1 (𝜑 → ((1st𝐼)‘𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106   I cid 5582  cres 5691  cfv 6563  1st c1st 8011  Basecbs 17245  Catccat 17709  idfunccidfu 17906
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-pow 5371  ax-pr 5438  ax-un 7754
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-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  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-1st 8013  df-idfu 17910
This theorem is referenced by:  idffth  17987  ressffth  17992  catciso  18165
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