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Theorem idfu1st 17859
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 2728 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
51, 2, 3, 4idfuval 17856 . . 3 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
65fveq2d 6896 . 2 (𝜑 → (1st𝐼) = (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩))
72fvexi 6906 . . . 4 𝐵 ∈ V
8 resiexg 7915 . . . 4 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
97, 8ax-mp 5 . . 3 ( I ↾ 𝐵) ∈ V
107, 7xpex 7750 . . . 4 (𝐵 × 𝐵) ∈ V
1110mptex 7230 . . 3 (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V
129, 11op1st 7996 . 2 (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩) = ( I ↾ 𝐵)
136, 12eqtrdi 2784 1 (𝜑 → (1st𝐼) = ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3470  cop 4631  cmpt 5226   I cid 5570   × cxp 5671  cres 5675  cfv 6543  1st c1st 7986  Basecbs 17174  Hom chom 17238  Catccat 17638  idfunccidfu 17835
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 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
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 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1st 7988  df-idfu 17839
This theorem is referenced by:  idfu1  17860  cofulid  17870  cofurid  17871  catciso  18094  curf2ndf  18233
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