Theorem zeroofn 18002

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 6889	. . 3 ⊢ (InitO‘𝑐) ∈ V
2	1	inex1 5287	. 2 ⊢ ((InitO‘𝑐) ∩ (TermO‘𝑐)) ∈ V
3		df-zeroo 17999	. 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
4	2, 3	fnmpti 6681	1 ⊢ ZeroO Fn Cat

Syntax hints:   ∩ cin 3925   Fn wfn 6526  ‘cfv 6531  Catccat 17676  InitOcinito 17994  TermOctermo 17995  ZeroOczeroo 17996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fn 6534  df-fv 6539  df-zeroo 17999

This theorem is referenced by:  zeroopropdlem  49159  zeroopropd  49162