Theorem zeroofn 18047

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 6896	. . 3 ⊢ (InitO‘𝑐) ∈ V
2	1	inex1 5287	. 2 ⊢ ((InitO‘𝑐) ∩ (TermO‘𝑐)) ∈ V
3		df-zeroo 18044	. 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
4	2, 3	fnmpti 6680	1 ⊢ ZeroO Fn Cat

Syntax hints:   ∩ cin 3905   Fn wfn 6533  ‘cfv 6538  Catccat 17721  InitOcinito 18039  TermOctermo 18040  ZeroOczeroo 18041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-zeroo 18044

This theorem is referenced by:  zeroopropdlem  50003  zeroopropd  50006