Theorem idfu1st 17788

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 2733	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5	1, 2, 3, 4	idfuval 17785	. . 3 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
6	5	fveq2d 6832	. 2 ⊢ (𝜑 → (1st ‘𝐼) = (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩))
7	2	fvexi 6842	. . . 4 ⊢ 𝐵 ∈ V
8		resiexg 7848	. . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
9	7, 8	ax-mp 5	. . 3 ⊢ ( I ↾ 𝐵) ∈ V
10	7, 7	xpex 7692	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
11	10	mptex 7163	. . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V
12	9, 11	op1st 7935	. 2 ⊢ (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩) = ( I ↾ 𝐵)
13	6, 12	eqtrdi 2784	1 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ⟨cop 4581   ↦ cmpt 5174   I cid 5513   × cxp 5617   ↾ cres 5621  ‘cfv 6486  1^st c1st 7925  Basecbs 17122  Hom chom 17174  Catccat 17572  id_funccidfu 17764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-1st 7927  df-idfu 17768

This theorem is referenced by:  idfu1  17789  cofulid  17799  cofurid  17800  catciso  18020  curf2ndf  18155  idfu1stf1o  49224  idfu1sta  49226