Theorem initopropd 49275

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 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
3		initopropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
5		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
6	2, 4, 5	initopropdlem 49272	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (InitO‘𝐶) = (InitO‘𝐷))
7	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
8	7	eqcomd 2737	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐷) = (Homf ‘𝐶))
9	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
10	9	eqcomd 2737	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐷) = (compf‘𝐶))
11		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → ¬ 𝐷 ∈ V)
12	8, 10, 11	initopropdlem 49272	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (InitO‘𝐷) = (InitO‘𝐶))
13	12	eqcomd 2737	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (InitO‘𝐶) = (InitO‘𝐷))
14	1	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (Homf ‘𝐶) = (Homf ‘𝐷))
15	14	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Homf ‘𝐶) = (Homf ‘𝐷))
16		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		eqidd 2732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐶))
19	15	homfeqbas 17597	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐷))
20	16, 17, 18, 19	homfeq 17595	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 3224	. . . . . . . . . . 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 3224	. . . . . . . . . 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 2817	. . . . . . . . 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 2581	. . . . . . . 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 3153	. . . . . . 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 579	. . . . . 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 2817	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (𝑎 ∈ (Base‘𝐶) ↔ 𝑎 ∈ (Base‘𝐷)))
29	19	raleqdv 3292	. . . . . . 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 632	. . . . . 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 279	. . . . 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 3397	. . . 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 484	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
34		eqid 2731	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
35	33, 34, 16	initoval 17895	. . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (compf‘𝐶) = (compf‘𝐷))
37		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐶 ∈ V)
38		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐷 ∈ V)
39	14, 36, 37, 38	catpropd 17610	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
40	39	biimpa 476	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
41		eqid 2731	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
42	40, 41, 17	initoval 17895	. . . 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 2776	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (InitO‘𝐶) = (InitO‘𝐷))
44	39	pm5.32i 574	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ↔ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat))
45	44, 43	sylbir 235	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat) → (InitO‘𝐶) = (InitO‘𝐷))
46		initofn 17889	. . . . . . . 8 ⊢ InitO Fn Cat
47	46	fndmi 6580	. . . . . . 7 ⊢ dom InitO = Cat
48	47	eleq2i 2823	. . . . . 6 ⊢ (𝐶 ∈ dom InitO ↔ 𝐶 ∈ Cat)
49		ndmfv 6849	. . . . . 6 ⊢ (¬ 𝐶 ∈ dom InitO → (InitO‘𝐶) = ∅)
50	48, 49	sylnbir 331	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (InitO‘𝐶) = ∅)
51	50	ad2antrl 728	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (InitO‘𝐶) = ∅)
52	47	eleq2i 2823	. . . . . 6 ⊢ (𝐷 ∈ dom InitO ↔ 𝐷 ∈ Cat)
53		ndmfv 6849	. . . . . 6 ⊢ (¬ 𝐷 ∈ dom InitO → (InitO‘𝐷) = ∅)
54	52, 53	sylnbir 331	. . . . 5 ⊢ (¬ 𝐷 ∈ Cat → (InitO‘𝐷) = ∅)
55	54	ad2antll 729	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (InitO‘𝐷) = ∅)
56	51, 55	eqtr4d 2769	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (InitO‘𝐶) = (InitO‘𝐷))
57	43, 45, 56	pm2.61ddan 813	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (InitO‘𝐶) = (InitO‘𝐷))
58	6, 13, 57	pm2.61dda 814	1 ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃!weu 2563  ∀wral 3047  {crab 3395  Vcvv 3436  ∅c0 4278  dom cdm 5611  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  Hom chom 17167  Catccat 17565  Hom_f chomf 17567  comp_fccomf 17568  InitOcinito 17883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-cat 17569  df-homf 17571  df-comf 17572  df-inito 17886

This theorem is referenced by:  zeroopropd  49277