Theorem termopropd 49206

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 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	termopropdlem 49203	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (TermO‘𝐶) = (TermO‘𝐷))
7	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
8	7	eqcomd 2735	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐷) = (Homf ‘𝐶))
9	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
10	9	eqcomd 2735	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐷) = (compf‘𝐶))
11		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → ¬ 𝐷 ∈ V)
12	8, 10, 11	termopropdlem 49203	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (TermO‘𝐷) = (TermO‘𝐶))
13	12	eqcomd 2735	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (TermO‘𝐶) = (TermO‘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 2729	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		eqidd 2730	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐶))
19	15	homfeqbas 17633	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐷))
20	16, 17, 18, 19	homfeq 17631	. . . . . . . . . . . . . 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 3263	. . . . . . . . . . . . . 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 287	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 3227	. . . . . . . . . . 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 3227	. . . . . . . . . 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 2814	. . . . . . . . 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 2579	. . . . . . . 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 3154	. . . . . . 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 579	. . . . . 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 2814	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (𝑎 ∈ (Base‘𝐶) ↔ 𝑎 ∈ (Base‘𝐷)))
31	19	raleqdv 3296	. . . . . . 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 632	. . . . . 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 279	. . . . 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 3404	. . . 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 484	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
36		eqid 2729	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
37	35, 36, 16	termoval 17932	. . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (compf‘𝐶) = (compf‘𝐷))
39		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐶 ∈ V)
40		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐷 ∈ V)
41	14, 38, 39, 40	catpropd 17646	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
42	41	biimpa 476	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
43		eqid 2729	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
44	42, 43, 17	termoval 17932	. . . 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 2774	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (TermO‘𝐶) = (TermO‘𝐷))
46	41	pm5.32i 574	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ↔ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat))
47	46, 45	sylbir 235	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat) → (TermO‘𝐶) = (TermO‘𝐷))
48		termofn 17926	. . . . . . . 8 ⊢ TermO Fn Cat
49	48	fndmi 6604	. . . . . . 7 ⊢ dom TermO = Cat
50	49	eleq2i 2820	. . . . . 6 ⊢ (𝐶 ∈ dom TermO ↔ 𝐶 ∈ Cat)
51		ndmfv 6875	. . . . . 6 ⊢ (¬ 𝐶 ∈ dom TermO → (TermO‘𝐶) = ∅)
52	50, 51	sylnbir 331	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (TermO‘𝐶) = ∅)
53	52	ad2antrl 728	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (TermO‘𝐶) = ∅)
54	49	eleq2i 2820	. . . . . 6 ⊢ (𝐷 ∈ dom TermO ↔ 𝐷 ∈ Cat)
55		ndmfv 6875	. . . . . 6 ⊢ (¬ 𝐷 ∈ dom TermO → (TermO‘𝐷) = ∅)
56	54, 55	sylnbir 331	. . . . 5 ⊢ (¬ 𝐷 ∈ Cat → (TermO‘𝐷) = ∅)
57	56	ad2antll 729	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (TermO‘𝐷) = ∅)
58	53, 57	eqtr4d 2767	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (TermO‘𝐶) = (TermO‘𝐷))
59	45, 47, 58	pm2.61ddan 813	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (TermO‘𝐶) = (TermO‘𝐷))
60	6, 13, 59	pm2.61dda 814	1 ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃!weu 2561  ∀wral 3044  {crab 3402  Vcvv 3444  ∅c0 4292  dom cdm 5631  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  Hom chom 17207  Catccat 17601  Hom_f chomf 17603  comp_fccomf 17604  TermOctermo 17920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-cat 17605  df-homf 17607  df-comf 17608  df-termo 17923

This theorem is referenced by:  zeroopropd  49207