Theorem termopropd 49906

Description: Two structures with the same base, hom-sets and composition operation have the same terminal objects. (Contributed by Zhi Wang, 26-Oct-2025.)

Hypotheses

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

Assertion

Ref	Expression
termopropd	⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))

Proof of Theorem termopropd

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 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
3		initopropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
4	3	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
5		simpr 489	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
6	2, 4, 5	termopropdlem 49903	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (TermO‘𝐶) = (TermO‘𝐷))
7	1	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
8	7	eqcomd 2775	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐷) = (Homf ‘𝐶))
9	3	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
10	9	eqcomd 2775	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐷) = (compf‘𝐶))
11		simpr 489	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → ¬ 𝐷 ∈ V)
12	8, 10, 11	termopropdlem 49903	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (TermO‘𝐷) = (TermO‘𝐶))
13	12	eqcomd 2775	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (TermO‘𝐶) = (TermO‘𝐷))
14	1	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (Homf ‘𝐶) = (Homf ‘𝐷))
15	14	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Homf ‘𝐶) = (Homf ‘𝐷))
16		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		eqidd 2770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐶))
19	15	homfeqbas 17751	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐷))
20	16, 17, 18, 19	homfeq 17749	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑏 ∈ (Base‘𝐶)∀𝑎 ∈ (Base‘𝐶)(𝑏(Hom ‘𝐶)𝑎) = (𝑏(Hom ‘𝐷)𝑎)))
21		ralcom 3299	. . . . . . . . . . . . . 14 ⊢ (∀𝑏 ∈ (Base‘𝐶)∀𝑎 ∈ (Base‘𝐶)(𝑏(Hom ‘𝐶)𝑎) = (𝑏(Hom ‘𝐷)𝑎) ↔ ∀𝑎 ∈ (Base‘𝐶)∀𝑏 ∈ (Base‘𝐶)(𝑏(Hom ‘𝐶)𝑎) = (𝑏(Hom ‘𝐷)𝑎))
22	20, 21	bitrdi 290	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑎 ∈ (Base‘𝐶)∀𝑏 ∈ (Base‘𝐶)(𝑏(Hom ‘𝐶)𝑎) = (𝑏(Hom ‘𝐷)𝑎)))
23	15, 22	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ∀𝑎 ∈ (Base‘𝐶)∀𝑏 ∈ (Base‘𝐶)(𝑏(Hom ‘𝐶)𝑎) = (𝑏(Hom ‘𝐷)𝑎))
24	23	r19.21bi 3263	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ∧ 𝑎 ∈ (Base‘𝐶)) → ∀𝑏 ∈ (Base‘𝐶)(𝑏(Hom ‘𝐶)𝑎) = (𝑏(Hom ‘𝐷)𝑎))
25	24	r19.21bi 3263	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (𝑏(Hom ‘𝐶)𝑎) = (𝑏(Hom ‘𝐷)𝑎))
26	25	eleq2d 2855	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑏(Hom ‘𝐶)𝑎) ↔ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)))
27	26	eubidv 2620	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ∧ 𝑎 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑎) ↔ ∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)))
28	27	ralbidva 3192	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ∧ 𝑎 ∈ (Base‘𝐶)) → (∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑎) ↔ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)))
29	28	pm5.32da 589	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((𝑎 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑎)) ↔ (𝑎 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎))))
30	19	eleq2d 2855	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (𝑎 ∈ (Base‘𝐶) ↔ 𝑎 ∈ (Base‘𝐷)))
31	19	raleqdv 3329	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎) ↔ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)))
32	30, 31	anbi12d 643	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((𝑎 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)) ↔ (𝑎 ∈ (Base‘𝐷) ∧ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎))))
33	29, 32	bitrd 282	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((𝑎 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑎)) ↔ (𝑎 ∈ (Base‘𝐷) ∧ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎))))
34	33	rabbidva2 3425	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → {𝑎 ∈ (Base‘𝐶) ∣ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑎)} = {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)})
35		simpr 489	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
36		eqid 2769	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
37	35, 36, 16	termoval 18050	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (TermO‘𝐶) = {𝑎 ∈ (Base‘𝐶) ∣ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑎)})
38	3	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (compf‘𝐶) = (compf‘𝐷))
39		simprl 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐶 ∈ V)
40		simprr 784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐷 ∈ V)
41	14, 38, 39, 40	catpropd 17764	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
42	41	biimpa 481	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
43		eqid 2769	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
44	42, 43, 17	termoval 18050	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (TermO‘𝐷) = {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)})
45	34, 37, 44	3eqtr4d 2814	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (TermO‘𝐶) = (TermO‘𝐷))
46	41	pm5.32i 584	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ↔ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat))
47	46, 45	sylbir 238	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat) → (TermO‘𝐶) = (TermO‘𝐷))
48		termofn 18044	. . . . . . . 8 ⊢ TermO Fn Cat
49	48	fndmi 6640	. . . . . . 7 ⊢ dom TermO = Cat
50	49	eleq2i 2861	. . . . . 6 ⊢ (𝐶 ∈ dom TermO ↔ 𝐶 ∈ Cat)
51		ndmfv 6914	. . . . . 6 ⊢ (¬ 𝐶 ∈ dom TermO → (TermO‘𝐶) = ∅)
52	50, 51	sylnbir 334	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (TermO‘𝐶) = ∅)
53	52	ad2antrl 740	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (TermO‘𝐶) = ∅)
54	49	eleq2i 2861	. . . . . 6 ⊢ (𝐷 ∈ dom TermO ↔ 𝐷 ∈ Cat)
55		ndmfv 6914	. . . . . 6 ⊢ (¬ 𝐷 ∈ dom TermO → (TermO‘𝐷) = ∅)
56	54, 55	sylnbir 334	. . . . 5 ⊢ (¬ 𝐷 ∈ Cat → (TermO‘𝐷) = ∅)
57	56	ad2antll 741	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (TermO‘𝐷) = ∅)
58	53, 57	eqtr4d 2807	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (TermO‘𝐶) = (TermO‘𝐷))
59	45, 47, 58	pm2.61ddan 825	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (TermO‘𝐶) = (TermO‘𝐷))
60	6, 13, 59	pm2.61dda 826	1 ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃!weu 2602  ∀wral 3085  {crab 3423  Vcvv 3463  ∅c0 4294  dom cdm 5662  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  Hom chom 17320  Catccat 17719  Hom_f chomf 17721  comp_fccomf 17722  TermOctermo 18038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-cat 17723  df-homf 17725  df-comf 17726  df-termo 18041

This theorem is referenced by:  zeroopropd  49907