Theorem idfu1st 17892

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 2735	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5	1, 2, 3, 4	idfuval 17889	. . 3 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
6	5	fveq2d 6880	. 2 ⊢ (𝜑 → (1st ‘𝐼) = (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩))
7	2	fvexi 6890	. . . 4 ⊢ 𝐵 ∈ V
8		resiexg 7908	. . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
9	7, 8	ax-mp 5	. . 3 ⊢ ( I ↾ 𝐵) ∈ V
10	7, 7	xpex 7747	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
11	10	mptex 7215	. . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V
12	9, 11	op1st 7996	. 2 ⊢ (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩) = ( I ↾ 𝐵)
13	6, 12	eqtrdi 2786	1 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607   ↦ cmpt 5201   I cid 5547   × cxp 5652   ↾ cres 5656  ‘cfv 6531  1^st c1st 7986  Basecbs 17228  Hom chom 17282  Catccat 17676  id_funccidfu 17868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-1st 7988  df-idfu 17872

This theorem is referenced by:  idfu1  17893  cofulid  17903  cofurid  17904  catciso  18124  curf2ndf  18259  idfu1sta  49060