Theorem idfu1 17925

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 17924	. . 3 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))
5	4	fveq1d 6908	. 2 ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = (( I ↾ 𝐵)‘𝑋))
6		idfu1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		fvresi 7193	. . 3 ⊢ (𝑋 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑋) = 𝑋)
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (( I ↾ 𝐵)‘𝑋) = 𝑋)
9	5, 8	eqtrd 2777	1 ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   I cid 5577   ↾ cres 5687  ‘cfv 6561  1^st c1st 8012  Basecbs 17247  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:  idffth  17980  ressffth  17985  catciso  18156