Theorem zeroopropdlem 50695

Description: Lemma for zeroopropd 50698. (Contributed by Zhi Wang, 26-Oct-2025.)

Hypotheses

Ref	Expression
initopropd.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
initopropd.2	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
initopropdlem.1	⊢ (𝜑 → ¬ 𝐶 ∈ V)

Assertion

Ref	Expression
zeroopropdlem	⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷))

Proof of Theorem zeroopropdlem

Step	Hyp	Ref	Expression
1		zeroofn 18214	. 2 ⊢ ZeroO Fn Cat
2		initopropdlem.1	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ V)
3		ssv 3937	. 2 ⊢ Cat ⊆ V
4		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat)
5		eqid 2763	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
6		eqid 2763	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7	4, 5, 6	zerooval 18220	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (ZeroO‘𝐷) = ((InitO‘𝐷) ∩ (TermO‘𝐷)))
8		initopropd.1	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
9		initopropd.2	. . . . . . . 8 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
10	8, 9, 2	initopropdlem 50693	. . . . . . 7 ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))
11		fvprc 6908	. . . . . . . 8 ⊢ (¬ 𝐶 ∈ V → (InitO‘𝐶) = ∅)
12	2, 11	syl 18	. . . . . . 7 ⊢ (𝜑 → (InitO‘𝐶) = ∅)
13	10, 12	eqtr3d 2800	. . . . . 6 ⊢ (𝜑 → (InitO‘𝐷) = ∅)
14	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (InitO‘𝐷) = ∅)
15	8, 9, 2	termopropdlem 50694	. . . . . . 7 ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))
16		fvprc 6908	. . . . . . . 8 ⊢ (¬ 𝐶 ∈ V → (TermO‘𝐶) = ∅)
17	2, 16	syl 18	. . . . . . 7 ⊢ (𝜑 → (TermO‘𝐶) = ∅)
18	15, 17	eqtr3d 2800	. . . . . 6 ⊢ (𝜑 → (TermO‘𝐷) = ∅)
19	18	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (TermO‘𝐷) = ∅)
20	14, 19	ineq12d 4121	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → ((InitO‘𝐷) ∩ (TermO‘𝐷)) = (∅ ∩ ∅))
21		inidm 4126	. . . 4 ⊢ (∅ ∩ ∅) = ∅
22	20, 21	eqtrdi 2814	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → ((InitO‘𝐷) ∩ (TermO‘𝐷)) = ∅)
23	7, 22	eqtrd 2798	. 2 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (ZeroO‘𝐷) = ∅)
24	1, 2, 3, 23	initopropdlemlem 50692	1 ⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092  ∅c0 4281  ‘cfv 6566  Basecbs 17431  Hom chom 17483  Catccat 17888  Hom_f chomf 17890  comp_fccomf 17891  InitOcinito 18206  TermOctermo 18207  ZeroOczeroo 18208

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-homf 17894  df-inito 18209  df-termo 18210  df-zeroo 18211

This theorem is used by:  zeroopropd  50698