Theorem zeroofn 17329

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 6676	. . 3 ⊢ (InitO‘𝑐) ∈ V
2	1	inex1 5191	. 2 ⊢ ((InitO‘𝑐) ∩ (TermO‘𝑐)) ∈ V
3		df-zeroo 17326	. 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
4	2, 3	fnmpti 6479	1 ⊢ ZeroO Fn Cat

Syntax hints:   ∩ cin 3859   Fn wfn 6335  ‘cfv 6340  Catccat 17007  InitOcinito 17321  TermOctermo 17322  ZeroOczeroo 17323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fn 6343  df-fv 6348  df-zeroo 17326

This theorem is referenced by: (None)