Theorem idfu2nd 17383

Description: Value of the morphism 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)
idfuval.h	⊢ 𝐻 = (Hom ‘𝐶)
idfu2nd.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
idfu2nd.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
idfu2nd	⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))

Proof of Theorem idfu2nd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7216	. 2 ⊢ (𝑋(2nd ‘𝐼)𝑌) = ((2nd ‘𝐼)‘⟨𝑋, 𝑌⟩)
2		idfuval.i	. . . . . 6 ⊢ 𝐼 = (idfunc‘𝐶)
3		idfuval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
4		idfuval.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		idfuval.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
6	2, 3, 4, 5	idfuval 17382	. . . . 5 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
7	6	fveq2d 6721	. . . 4 ⊢ (𝜑 → (2nd ‘𝐼) = (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩))
8	3	fvexi 6731	. . . . . 6 ⊢ 𝐵 ∈ V
9		resiexg 7692	. . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
10	8, 9	ax-mp 5	. . . . 5 ⊢ ( I ↾ 𝐵) ∈ V
11	8, 8	xpex 7538	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
12	11	mptex 7039	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V
13	10, 12	op2nd 7770	. . . 4 ⊢ (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))
14	7, 13	eqtrdi 2794	. . 3 ⊢ (𝜑 → (2nd ‘𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))))
15		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
16	15	fveq2d 6721	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
17		df-ov 7216	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
18	16, 17	eqtr4di 2796	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝑋𝐻𝑌))
19	18	reseq2d 5851	. . 3 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ( I ↾ (𝐻‘𝑧)) = ( I ↾ (𝑋𝐻𝑌)))
20		idfu2nd.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
21		idfu2nd.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
22	20, 21	opelxpd 5589	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
23		ovex 7246	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
24		resiexg 7692	. . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
25	23, 24	mp1i 13	. . 3 ⊢ (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
26	14, 19, 22, 25	fvmptd 6825	. 2 ⊢ (𝜑 → ((2nd ‘𝐼)‘⟨𝑋, 𝑌⟩) = ( I ↾ (𝑋𝐻𝑌)))
27	1, 26	syl5eq 2790	1 ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  Vcvv 3408  ⟨cop 4547   ↦ cmpt 5135   I cid 5454   × cxp 5549   ↾ cres 5553  ‘cfv 6380  (class class class)co 7213  2^nd c2nd 7760  Basecbs 16760  Hom chom 16813  Catccat 17167  id_funccidfu 17361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-2nd 7762  df-idfu 17365

This theorem is referenced by:  idfu2  17384  idfucl  17387  cofulid  17396  cofurid  17397  idffth  17440  ressffth  17445  catciso  17617