Theorem initofn 17952

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 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6847	. . 3 ⊢ (Base‘𝑐) ∈ V
2	1	rabex 5274	. 2 ⊢ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)} ∈ V
3		df-inito 17949	. 2 ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
4	2, 3	fnmpti 6635	1 ⊢ InitO Fn Cat

Syntax hints:   ∈ wcel 2119  ∃!weu 2572  ∀wral 3054  {crab 3392   Fn wfn 6487  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  Hom chom 17229  Catccat 17628  InitOcinito 17946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-inito 17949

This theorem is referenced by:  dftermo3  17971  initopropdlem  49737  initopropd  49740  dfinito4  49998