Theorem idfu1st 17924

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 2737	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5	1, 2, 3, 4	idfuval 17921	. . 3 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
6	5	fveq2d 6910	. 2 ⊢ (𝜑 → (1st ‘𝐼) = (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩))
7	2	fvexi 6920	. . . 4 ⊢ 𝐵 ∈ V
8		resiexg 7934	. . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
9	7, 8	ax-mp 5	. . 3 ⊢ ( I ↾ 𝐵) ∈ V
10	7, 7	xpex 7773	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
11	10	mptex 7243	. . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V
12	9, 11	op1st 8022	. 2 ⊢ (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩) = ( I ↾ 𝐵)
13	6, 12	eqtrdi 2793	1 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ⟨cop 4632   ↦ cmpt 5225   I cid 5577   × cxp 5683   ↾ cres 5687  ‘cfv 6561  1^st c1st 8012  Basecbs 17247  Hom chom 17308  Catccat 17707  id_funccidfu 17900

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-idfu 17904

This theorem is referenced by:  idfu1  17925  cofulid  17935  cofurid  17936  catciso  18156  curf2ndf  18292