Theorem idfu1st 17846

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 2736	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5	1, 2, 3, 4	idfuval 17843	. . 3 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
6	5	fveq2d 6844	. 2 ⊢ (𝜑 → (1st ‘𝐼) = (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩))
7	2	fvexi 6854	. . . 4 ⊢ 𝐵 ∈ V
8		resiexg 7863	. . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
9	7, 8	ax-mp 5	. . 3 ⊢ ( I ↾ 𝐵) ∈ V
10	7, 7	xpex 7707	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
11	10	mptex 7178	. . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V
12	9, 11	op1st 7950	. 2 ⊢ (1st ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩) = ( I ↾ 𝐵)
13	6, 12	eqtrdi 2787	1 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573   ↦ cmpt 5166   I cid 5525   × cxp 5629   ↾ cres 5633  ‘cfv 6498  1^st c1st 7940  Basecbs 17179  Hom chom 17231  Catccat 17630  id_funccidfu 17822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-1st 7942  df-idfu 17826

This theorem is referenced by:  idfu1  17847  cofulid  17857  cofurid  17858  catciso  18078  curf2ndf  18213  idfu1stf1o  49574  idfu1sta  49576