Theorem initopropdlem 50075

Description: Lemma for initopropd 50078. (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
initopropdlem	⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))

Proof of Theorem initopropdlem

Dummy variables 𝑎 𝑏 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		initofn 18066	. 2 ⊢ InitO Fn Cat
2		initopropdlem.1	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ V)
3		ssv 3962	. 2 ⊢ Cat ⊆ V
4		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat)
5		eqid 2765	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
6		eqid 2765	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7	4, 5, 6	initoval 18072	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (InitO‘𝐷) = {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)})
8		initopropd.1	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
9		fvprc 6877	. . . . . . . . 9 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅)
10	2, 9	syl 18	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐶) = ∅)
11	8, 10	eqtr3d 2802	. . . . . . 7 ⊢ (𝜑 → (Homf ‘𝐷) = ∅)
12		homf0 49844	. . . . . . 7 ⊢ ((Base‘𝐷) = ∅ ↔ (Homf ‘𝐷) = ∅)
13	11, 12	sylibr 237	. . . . . 6 ⊢ (𝜑 → (Base‘𝐷) = ∅)
14	13	rabeqdv 3433	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)} = {𝑎 ∈ ∅ ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)})
15		rab0 4342	. . . . 5 ⊢ {𝑎 ∈ ∅ ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)} = ∅
16	14, 15	eqtrdi 2816	. . . 4 ⊢ (𝜑 → {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)} = ∅)
17	16	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)} = ∅)
18	7, 17	eqtrd 2800	. 2 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (InitO‘𝐷) = ∅)
19	1, 2, 3, 18	initopropdlemlem 50074	1 ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃!weu 2598  ∀wral 3081  {crab 3418  Vcvv 3457  ∅c0 4286  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  Hom chom 17343  Catccat 17742  Hom_f chomf 17744  comp_fccomf 17745  InitOcinito 18060

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-homf 17748  df-inito 18063

This theorem is used by:  zeroopropdlem  50077  initopropd  50078