Theorem initopropd 50169

Description: Two structures with the same base, hom-sets and composition operation have the same initial objects. (Contributed by Zhi Wang, 23-Oct-2025.)

Hypotheses

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

Assertion

Ref	Expression
initopropd	⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))

Proof of Theorem initopropd

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

Step	Hyp	Ref	Expression
1		initopropd.1	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
3		initopropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
4	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
5		simpr 490	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
6	2, 4, 5	initopropdlem 50166	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (InitO‘𝐶) = (InitO‘𝐷))
7	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
8	7	eqcomd 2766	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐷) = (Homf ‘𝐶))
9	3	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
10	9	eqcomd 2766	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐷) = (compf‘𝐶))
11		simpr 490	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → ¬ 𝐷 ∈ V)
12	8, 10, 11	initopropdlem 50166	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (InitO‘𝐷) = (InitO‘𝐶))
13	12	eqcomd 2766	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (InitO‘𝐶) = (InitO‘𝐷))
14	1	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (Homf ‘𝐶) = (Homf ‘𝐷))
15	14	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Homf ‘𝐶) = (Homf ‘𝐷))
16		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		eqidd 2761	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐶))
19	15	homfeqbas 17784	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐷))
20	16, 17, 18, 19	homfeq 17782	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑎 ∈ (Base‘𝐶)∀𝑏 ∈ (Base‘𝐶)(𝑎(Hom ‘𝐶)𝑏) = (𝑎(Hom ‘𝐷)𝑏)))
21	15, 20	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ∀𝑎 ∈ (Base‘𝐶)∀𝑏 ∈ (Base‘𝐶)(𝑎(Hom ‘𝐶)𝑏) = (𝑎(Hom ‘𝐷)𝑏))
22	21	r19.21bi 3254	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ∧ 𝑎 ∈ (Base‘𝐶)) → ∀𝑏 ∈ (Base‘𝐶)(𝑎(Hom ‘𝐶)𝑏) = (𝑎(Hom ‘𝐷)𝑏))
23	22	r19.21bi 3254	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (𝑎(Hom ‘𝐶)𝑏) = (𝑎(Hom ‘𝐷)𝑏))
24	23	eleq2d 2846	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑎(Hom ‘𝐶)𝑏) ↔ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)))
25	24	eubidv 2611	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐶)𝑏) ↔ ∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)))
26	25	ralbidva 3183	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ∧ 𝑎 ∈ (Base‘𝐶)) → (∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐶)𝑏) ↔ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)))
27	26	pm5.32da 590	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((𝑎 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐶)𝑏)) ↔ (𝑎 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏))))
28	19	eleq2d 2846	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (𝑎 ∈ (Base‘𝐶) ↔ 𝑎 ∈ (Base‘𝐷)))
29	19	raleqdv 3319	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏) ↔ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)))
30	28, 29	anbi12d 644	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((𝑎 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)) ↔ (𝑎 ∈ (Base‘𝐷) ∧ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏))))
31	27, 30	bitrd 282	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((𝑎 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐶)𝑏)) ↔ (𝑎 ∈ (Base‘𝐷) ∧ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏))))
32	31	rabbidva2 3414	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → {𝑎 ∈ (Base‘𝐶) ∣ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐶)𝑏)} = {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)})
33		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
34		eqid 2760	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
35	33, 34, 16	initoval 18082	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (InitO‘𝐶) = {𝑎 ∈ (Base‘𝐶) ∣ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐶)𝑏)})
36	3	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (compf‘𝐶) = (compf‘𝐷))
37		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐶 ∈ V)
38		simprr 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐷 ∈ V)
39	14, 36, 37, 38	catpropd 17797	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
40	39	biimpa 482	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
41		eqid 2760	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
42	40, 41, 17	initoval 18082	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (InitO‘𝐷) = {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝐷)𝑏)})
43	32, 35, 42	3eqtr4d 2805	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (InitO‘𝐶) = (InitO‘𝐷))
44	39	pm5.32i 585	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ↔ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat))
45	44, 43	sylbir 238	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat) → (InitO‘𝐶) = (InitO‘𝐷))
46		initofn 18076	. . . . . . . 8 ⊢ InitO Fn Cat
47	46	fndmi 6636	. . . . . . 7 ⊢ dom InitO = Cat
48	47	eleq2i 2852	. . . . . 6 ⊢ (𝐶 ∈ dom InitO ↔ 𝐶 ∈ Cat)
49		ndmfv 6910	. . . . . 6 ⊢ (¬ 𝐶 ∈ dom InitO → (InitO‘𝐶) = ∅)
50	48, 49	sylnbir 334	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (InitO‘𝐶) = ∅)
51	50	ad2antrl 741	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (InitO‘𝐶) = ∅)
52	47	eleq2i 2852	. . . . . 6 ⊢ (𝐷 ∈ dom InitO ↔ 𝐷 ∈ Cat)
53		ndmfv 6910	. . . . . 6 ⊢ (¬ 𝐷 ∈ dom InitO → (InitO‘𝐷) = ∅)
54	52, 53	sylnbir 334	. . . . 5 ⊢ (¬ 𝐷 ∈ Cat → (InitO‘𝐷) = ∅)
55	54	ad2antll 742	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (InitO‘𝐷) = ∅)
56	51, 55	eqtr4d 2798	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (InitO‘𝐶) = (InitO‘𝐷))
57	43, 45, 56	pm2.61ddan 826	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (InitO‘𝐶) = (InitO‘𝐷))
58	6, 13, 57	pm2.61dda 827	1 ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃!weu 2593  ∀wral 3076  {crab 3412  Vcvv 3450  ∅c0 4279  dom cdm 5655  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  Hom chom 17353  Catccat 17752  Hom_f chomf 17754  comp_fccomf 17755  InitOcinito 18070

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-cat 17756  df-homf 17758  df-comf 17759  df-inito 18073

This theorem is used by:  zeroopropd  50171