Theorem idfu2nd 17801

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 7361	. 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 17800	. . . . 5 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
7	6	fveq2d 6838	. . . 4 ⊢ (𝜑 → (2nd ‘𝐼) = (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩))
8	3	fvexi 6848	. . . . . 6 ⊢ 𝐵 ∈ V
9		resiexg 7854	. . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
10	8, 9	ax-mp 5	. . . . 5 ⊢ ( I ↾ 𝐵) ∈ V
11	8, 8	xpex 7698	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
12	11	mptex 7169	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V
13	10, 12	op2nd 7942	. . . 4 ⊢ (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))
14	7, 13	eqtrdi 2787	. . 3 ⊢ (𝜑 → (2nd ‘𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))))
15		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
16	15	fveq2d 6838	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
17		df-ov 7361	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
18	16, 17	eqtr4di 2789	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝑋𝐻𝑌))
19	18	reseq2d 5938	. . 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 5663	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
23		ovex 7391	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
24		resiexg 7854	. . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
25	23, 24	mp1i 13	. . 3 ⊢ (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
26	14, 19, 22, 25	fvmptd 6948	. 2 ⊢ (𝜑 → ((2nd ‘𝐼)‘⟨𝑋, 𝑌⟩) = ( I ↾ (𝑋𝐻𝑌)))
27	1, 26	eqtrid 2783	1 ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ⟨cop 4586   ↦ cmpt 5179   I cid 5518   × cxp 5622   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358  2^nd c2nd 7932  Basecbs 17136  Hom chom 17188  Catccat 17587  id_funccidfu 17779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-2nd 7934  df-idfu 17783

This theorem is referenced by:  idfu2  17802  idfucl  17805  cofulid  17814  cofurid  17815  idffth  17859  ressffth  17864  catciso  18035  idfu2nda  49344  cofidf2a  49358