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Theorem idfu1 16746
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 16745 . . 3 (𝜑 → (1st𝐼) = ( I ↾ 𝐵))
54fveq1d 6334 . 2 (𝜑 → ((1st𝐼)‘𝑋) = (( I ↾ 𝐵)‘𝑋))
6 idfu1.x . . 3 (𝜑𝑋𝐵)
7 fvresi 6582 . . 3 (𝑋𝐵 → (( I ↾ 𝐵)‘𝑋) = 𝑋)
86, 7syl 17 . 2 (𝜑 → (( I ↾ 𝐵)‘𝑋) = 𝑋)
95, 8eqtrd 2805 1 (𝜑 → ((1st𝐼)‘𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145   I cid 5156  cres 5251  cfv 6031  1st c1st 7312  Basecbs 16063  Catccat 16531  idfunccidfu 16721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1st 7314  df-idfu 16725
This theorem is referenced by:  idffth  16799  ressffth  16804  catciso  16963
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