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Theorem idfu1st 17773
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 2733 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
51, 2, 3, 4idfuval 17770 . . 3 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
65fveq2d 6850 . 2 (𝜑 → (1st𝐼) = (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩))
72fvexi 6860 . . . 4 𝐵 ∈ V
8 resiexg 7855 . . . 4 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
97, 8ax-mp 5 . . 3 ( I ↾ 𝐵) ∈ V
107, 7xpex 7691 . . . 4 (𝐵 × 𝐵) ∈ V
1110mptex 7177 . . 3 (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V
129, 11op1st 7933 . 2 (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩) = ( I ↾ 𝐵)
136, 12eqtrdi 2789 1 (𝜑 → (1st𝐼) = ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3447  cop 4596  cmpt 5192   I cid 5534   × cxp 5635  cres 5639  cfv 6500  1st c1st 7923  Basecbs 17091  Hom chom 17152  Catccat 17552  idfunccidfu 17749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1st 7925  df-idfu 17753
This theorem is referenced by:  idfu1  17774  cofulid  17784  cofurid  17785  catciso  18005  curf2ndf  18144
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