Theorem initopropdlem 50469

Description: Lemma for initopropd 50472. (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 18169	. 2 ⊢ InitO Fn Cat
2		initopropdlem.1	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ V)
3		ssv 3933	. 2 ⊢ Cat ⊆ V
4		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat)
5		eqid 2760	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
6		eqid 2760	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7	4, 5, 6	initoval 18175	. . 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 6880	. . . . . . . . 9 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅)
10	2, 9	syl 18	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐶) = ∅)
11	8, 10	eqtr3d 2797	. . . . . . 7 ⊢ (𝜑 → (Homf ‘𝐷) = ∅)
12		homf0 50230	. . . . . . 7 ⊢ ((Base‘𝐷) = ∅ ↔ (Homf ‘𝐷) = ∅)
13	11, 12	sylibr 237	. . . . . 6 ⊢ (𝜑 → (Base‘𝐷) = ∅)
14	13	rabeqdv 3427	. . . . 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 4334	. . . . 5 ⊢ {𝑎 ∈ ∅ ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)} = ∅
16	14, 15	eqtrdi 2811	. . . 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 2795	. 2 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (InitO‘𝐷) = ∅)
19	1, 2, 3, 18	initopropdlemlem 50468	1 ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃!weu 2593  ∀wral 3076  {crab 3412  Vcvv 3450  ∅c0 4277  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  Hom chom 17446  Catccat 17845  Hom_f chomf 17847  comp_fccomf 17848  InitOcinito 18163

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-homf 17851  df-inito 18166

This theorem is used by:  zeroopropdlem  50471  initopropd  50472