Theorem zeroofn 18043

Description: ZeroO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)

Assertion

Ref	Expression
zeroofn	⊢ ZeroO Fn Cat

Proof of Theorem zeroofn

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6920	. . 3 ⊢ (InitO‘𝑐) ∈ V
2	1	inex1 5323	. 2 ⊢ ((InitO‘𝑐) ∩ (TermO‘𝑐)) ∈ V
3		df-zeroo 18040	. 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
4	2, 3	fnmpti 6712	1 ⊢ ZeroO Fn Cat

Syntax hints:   ∩ cin 3962   Fn wfn 6558  ‘cfv 6563  Catccat 17709  InitOcinito 18035  TermOctermo 18036  ZeroOczeroo 18037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-zeroo 18040

This theorem is referenced by: (None)