MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  idfu2nd Structured version   Visualization version   GIF version

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 = (idfunc𝐶)
idfuval.b 𝐵 = (Base‘𝐶)
idfuval.c (𝜑𝐶 ∈ Cat)
idfuval.h 𝐻 = (Hom ‘𝐶)
idfu2nd.x (𝜑𝑋𝐵)
idfu2nd.y (𝜑𝑌𝐵)
Assertion
Ref Expression
idfu2nd (𝜑 → (𝑋(2nd𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))

Proof of Theorem idfu2nd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ov 7216 . 2 (𝑋(2nd𝐼)𝑌) = ((2nd𝐼)‘⟨𝑋, 𝑌⟩)
2 idfuval.i . . . . . 6 𝐼 = (idfunc𝐶)
3 idfuval.b . . . . . 6 𝐵 = (Base‘𝐶)
4 idfuval.c . . . . . 6 (𝜑𝐶 ∈ Cat)
5 idfuval.h . . . . . 6 𝐻 = (Hom ‘𝐶)
62, 3, 4, 5idfuval 17382 . . . . 5 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
76fveq2d 6721 . . . 4 (𝜑 → (2nd𝐼) = (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩))
83fvexi 6731 . . . . . 6 𝐵 ∈ V
9 resiexg 7692 . . . . . 6 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
108, 9ax-mp 5 . . . . 5 ( I ↾ 𝐵) ∈ V
118, 8xpex 7538 . . . . . 6 (𝐵 × 𝐵) ∈ V
1211mptex 7039 . . . . 5 (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))) ∈ V
1310, 12op2nd 7770 . . . 4 (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))
147, 13eqtrdi 2794 . . 3 (𝜑 → (2nd𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))))
15 simpr 488 . . . . . 6 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
1615fveq2d 6721 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
17 df-ov 7216 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1816, 17eqtr4di 2796 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝑋𝐻𝑌))
1918reseq2d 5851 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ( I ↾ (𝐻𝑧)) = ( I ↾ (𝑋𝐻𝑌)))
20 idfu2nd.x . . . 4 (𝜑𝑋𝐵)
21 idfu2nd.y . . . 4 (𝜑𝑌𝐵)
2220, 21opelxpd 5589 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
23 ovex 7246 . . . 4 (𝑋𝐻𝑌) ∈ V
24 resiexg 7692 . . . 4 ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
2523, 24mp1i 13 . . 3 (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
2614, 19, 22, 25fvmptd 6825 . 2 (𝜑 → ((2nd𝐼)‘⟨𝑋, 𝑌⟩) = ( I ↾ (𝑋𝐻𝑌)))
271, 26syl5eq 2790 1 (𝜑 → (𝑋(2nd𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
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  2nd c2nd 7760  Basecbs 16760  Hom chom 16813  Catccat 17167  idfunccidfu 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
  Copyright terms: Public domain W3C validator