Theorem idfu2nd 17802

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 7356	. 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 17801	. . . . 5 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
7	6	fveq2d 6830	. . . 4 ⊢ (𝜑 → (2nd ‘𝐼) = (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩))
8	3	fvexi 6840	. . . . . 6 ⊢ 𝐵 ∈ V
9		resiexg 7852	. . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
10	8, 9	ax-mp 5	. . . . 5 ⊢ ( I ↾ 𝐵) ∈ V
11	8, 8	xpex 7693	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
12	11	mptex 7163	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V
13	10, 12	op2nd 7940	. . . 4 ⊢ (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))
14	7, 13	eqtrdi 2780	. . 3 ⊢ (𝜑 → (2nd ‘𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))))
15		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
16	15	fveq2d 6830	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
17		df-ov 7356	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
18	16, 17	eqtr4di 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝑋𝐻𝑌))
19	18	reseq2d 5934	. . 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 5662	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
23		ovex 7386	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
24		resiexg 7852	. . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
25	23, 24	mp1i 13	. . 3 ⊢ (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
26	14, 19, 22, 25	fvmptd 6941	. 2 ⊢ (𝜑 → ((2nd ‘𝐼)‘⟨𝑋, 𝑌⟩) = ( I ↾ (𝑋𝐻𝑌)))
27	1, 26	eqtrid 2776	1 ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ⟨cop 4585   ↦ cmpt 5176   I cid 5517   × cxp 5621   ↾ cres 5625  ‘cfv 6486  (class class class)co 7353  2^nd c2nd 7930  Basecbs 17138  Hom chom 17190  Catccat 17588  id_funccidfu 17780

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-2nd 7932  df-idfu 17784

This theorem is referenced by:  idfu2  17803  idfucl  17806  cofulid  17815  cofurid  17816  idffth  17860  ressffth  17865  catciso  18036  idfu2nda  49089  cofidf2a  49103