Theorem idfu1st 17958

Description: Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
idfuval.i	⊢ 𝐼 = (idfunc‘𝐶)
idfuval.b	⊢ 𝐵 = (Base‘𝐶)
idfuval.c	⊢ (𝜑 → 𝐶 ∈ Cat)

Assertion

Ref	Expression
idfu1st	⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))

Proof of Theorem idfu1st

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idfuval.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐶)
2		idfuval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		idfuval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		eqid 2765	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5	1, 2, 3, 4	idfuval 17955	. . 3 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
6	5	fveq2d 6889	. 2 ⊢ (𝜑 → (1st ‘𝐼) = (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩))
7	2	fvexi 6899	. . . 4 ⊢ 𝐵 ∈ V
8		resiexg 7915	. . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
9	7, 8	ax-mp 5	. . 3 ⊢ ( I ↾ 𝐵) ∈ V
10	7, 7	xpex 7758	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
11	10	mptex 7228	. . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V
12	9, 11	op1st 8000	. 2 ⊢ (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩) = ( I ↾ 𝐵)
13	6, 12	eqtrdi 2816	1 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597   ↦ cmpt 5194   I cid 5557   × cxp 5661   ↾ cres 5665  ‘cfv 6540  1^st c1st 7990  Basecbs 17291  Hom chom 17343  Catccat 17742  id_funccidfu 17934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1st 7992  df-idfu 17938

This theorem is used by:  idfu1  17959  cofulid  17969  cofurid  17970  catciso  18190  curf2ndf  18325  idfu1stf1o  49934  idfu1sta  49936