Theorem zeroofn 17893

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 6835	. . 3 ⊢ (InitO‘𝑐) ∈ V
2	1	inex1 5255	. 2 ⊢ ((InitO‘𝑐) ∩ (TermO‘𝑐)) ∈ V
3		df-zeroo 17890	. 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
4	2, 3	fnmpti 6624	1 ⊢ ZeroO Fn Cat

Syntax hints:   ∩ cin 3901   Fn wfn 6476  ‘cfv 6481  Catccat 17567  InitOcinito 17885  TermOctermo 17886  ZeroOczeroo 17887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489  df-zeroo 17890

This theorem is referenced by:  zeroopropdlem  49273  zeroopropd  49276