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Theorem idfu1st 17221
 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)
Assertion
Ref Expression
idfu1st (𝜑 → (1st𝐼) = ( I ↾ 𝐵))

Proof of Theorem idfu1st
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 idfuval.i . . . 4 𝐼 = (idfunc𝐶)
2 idfuval.b . . . 4 𝐵 = (Base‘𝐶)
3 idfuval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 eqid 2758 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
51, 2, 3, 4idfuval 17218 . . 3 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
65fveq2d 6667 . 2 (𝜑 → (1st𝐼) = (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩))
72fvexi 6677 . . . 4 𝐵 ∈ V
8 resiexg 7630 . . . 4 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
97, 8ax-mp 5 . . 3 ( I ↾ 𝐵) ∈ V
107, 7xpex 7480 . . . 4 (𝐵 × 𝐵) ∈ V
1110mptex 6983 . . 3 (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V
129, 11op1st 7707 . 2 (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩) = ( I ↾ 𝐵)
136, 12eqtrdi 2809 1 (𝜑 → (1st𝐼) = ( I ↾ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4531   ↦ cmpt 5116   I cid 5433   × cxp 5526   ↾ cres 5530  ‘cfv 6340  1st c1st 7697  Basecbs 16554  Hom chom 16647  Catccat 17006  idfunccidfu 17197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-1st 7699  df-idfu 17201 This theorem is referenced by:  idfu1  17222  cofulid  17232  cofurid  17233  catciso  17446  curf2ndf  17576
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