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

Theorem initofn 17983
Description: InitO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
Assertion
Ref Expression
initofn InitO Fn Cat

Proof of Theorem initofn
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6915 . . 3 (Base‘𝑐) ∈ V
21rabex 5338 . 2 {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)} ∈ V
3 df-inito 17980 . 2 InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)})
42, 3fnmpti 6703 1 InitO Fn Cat
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  ∃!weu 2557  wral 3058  {crab 3430   Fn wfn 6548  cfv 6553  (class class class)co 7426  Basecbs 17187  Hom chom 17251  Catccat 17651  InitOcinito 17977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561  df-inito 17980
This theorem is referenced by:  dftermo3  18002
  Copyright terms: Public domain W3C validator