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Theorem idfu1 17142
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 17141 . . 3 (𝜑 → (1st𝐼) = ( I ↾ 𝐵))
54fveq1d 6668 . 2 (𝜑 → ((1st𝐼)‘𝑋) = (( I ↾ 𝐵)‘𝑋))
6 idfu1.x . . 3 (𝜑𝑋𝐵)
7 fvresi 6930 . . 3 (𝑋𝐵 → (( I ↾ 𝐵)‘𝑋) = 𝑋)
86, 7syl 17 . 2 (𝜑 → (( I ↾ 𝐵)‘𝑋) = 𝑋)
95, 8eqtrd 2860 1 (𝜑 → ((1st𝐼)‘𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2106   I cid 5457  cres 5555  cfv 6351  1st c1st 7681  Basecbs 16475  Catccat 16927  idfunccidfu 17117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-1st 7683  df-idfu 17121
This theorem is referenced by:  idffth  17195  ressffth  17200  catciso  17359
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