Theorem initofn 17730

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 6805	. . 3 ⊢ (Base‘𝑐) ∈ V
2	1	rabex 5259	. 2 ⊢ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)} ∈ V
3		df-inito 17727	. 2 ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
4	2, 3	fnmpti 6594	1 ⊢ InitO Fn Cat

Syntax hints:   ∈ wcel 2101  ∃!weu 2563  ∀wral 3059  {crab 3221   Fn wfn 6442  ‘cfv 6447  (class class class)co 7295  Basecbs 16940  Hom chom 17001  Catccat 17401  InitOcinito 17724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-iota 6399  df-fun 6449  df-fn 6450  df-fv 6455  df-inito 17727

This theorem is referenced by:  dftermo3  17749