Theorem zeroopropdlem 50020

Description: Lemma for zeroopropd 50023. (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 18041	. 2 ⊢ ZeroO Fn Cat
2		initopropdlem.1	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ V)
3		ssv 3961	. 2 ⊢ Cat ⊆ V
4		simpr 489	. . . 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 18047	. . 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 50018	. . . . . . 7 ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))
11		fvprc 6873	. . . . . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (InitO‘𝐷) = ∅)
15	8, 9, 2	termopropdlem 50019	. . . . . . 7 ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))
16		fvprc 6873	. . . . . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (TermO‘𝐷) = ∅)
20	14, 19	ineq12d 4174	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → ((InitO‘𝐷) ∩ (TermO‘𝐷)) = (∅ ∩ ∅))
21		inidm 4179	. . . 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 50017	1 ⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∩ cin 3904  ∅c0 4286  ‘cfv 6536  Basecbs 17264  Hom chom 17316  Catccat 17715  Hom_f chomf 17717  comp_fccomf 17718  InitOcinito 18033  TermOctermo 18034  ZeroOczeroo 18035

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-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

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-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-homf 17721  df-inito 18036  df-termo 18037  df-zeroo 18038

This theorem is referenced by:  zeroopropd  50023