Theorem idfu1 17865

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)
idfu1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
idfu1	⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)

Proof of Theorem idfu1

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	1, 2, 3	idfu1st 17864	. . 3 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))
5	4	fveq1d 6899	. 2 ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = (( I ↾ 𝐵)‘𝑋))
6		idfu1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		fvresi 7182	. . 3 ⊢ (𝑋 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑋) = 𝑋)
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (( I ↾ 𝐵)‘𝑋) = 𝑋)
9	5, 8	eqtrd 2768	1 ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   I cid 5575   ↾ cres 5680  ‘cfv 6548  1^st c1st 7991  Basecbs 17179  Catccat 17643  id_funccidfu 17840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-1st 7993  df-idfu 17844

This theorem is referenced by:  idffth  17921  ressffth  17926  catciso  18099