Theorem idfu2nd 17935

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 7415	. 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 17934	. . . . 5 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
7	6	fveq2d 6887	. . . 4 ⊢ (𝜑 → (2nd ‘𝐼) = (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩))
8	3	fvexi 6897	. . . . . 6 ⊢ 𝐵 ∈ V
9		resiexg 7910	. . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
10	8, 9	ax-mp 5	. . . . 5 ⊢ ( I ↾ 𝐵) ∈ V
11	8, 8	xpex 7753	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
12	11	mptex 7223	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V
13	10, 12	op2nd 7996	. . . 4 ⊢ (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))
14	7, 13	eqtrdi 2814	. . 3 ⊢ (𝜑 → (2nd ‘𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))))
15		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
16	15	fveq2d 6887	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
17		df-ov 7415	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
18	16, 17	eqtr4di 2816	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝑋𝐻𝑌))
19	18	reseq2d 5980	. . 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 5702	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
23		ovex 7445	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
24		resiexg 7910	. . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
25	23, 24	mp1i 14	. . 3 ⊢ (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
26	14, 19, 22, 25	fvmptd 6999	. 2 ⊢ (𝜑 → ((2nd ‘𝐼)‘⟨𝑋, 𝑌⟩) = ( I ↾ (𝑋𝐻𝑌)))
27	1, 26	eqtrid 2810	1 ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4596   ↦ cmpt 5193   I cid 5557   × cxp 5661   ↾ cres 5665  ‘cfv 6538  (class class class)co 7412  2^nd c2nd 7986  Basecbs 17270  Hom chom 17322  Catccat 17721  id_funccidfu 17913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-2nd 7988  df-idfu 17917

This theorem is referenced by:  idfu2  17936  idfucl  17939  cofulid  17948  cofurid  17949  idffth  17993  ressffth  17998  catciso  18169  idfu2nda  49864  cofidf2a  49878