Theorem termopropd 49229

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 49226	. 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 49226	. . 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 17602	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐷))
20	16, 17, 18, 19	homfeq 17600	. . . . . . . . . . . . . 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 3257	. . . . . . . . . . . . . 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 3221	. . . . . . . . . . 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 3221	. . . . . . . . . 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 3150	. . . . . . 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 3289	. . . . . . 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 3396	. . . 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 17901	. . . 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 17615	. . . . . 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 17901	. . . 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 17895	. . . . . . . 8 ⊢ TermO Fn Cat
49	48	fndmi 6586	. . . . . . 7 ⊢ dom TermO = Cat
50	49	eleq2i 2820	. . . . . 6 ⊢ (𝐶 ∈ dom TermO ↔ 𝐶 ∈ Cat)
51		ndmfv 6855	. . . . . 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 6855	. . . . . 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 3394  Vcvv 3436  ∅c0 4284  dom cdm 5619  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  Hom chom 17172  Catccat 17570  Hom_f chomf 17572  comp_fccomf 17573  TermOctermo 17889

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-cat 17574  df-homf 17576  df-comf 17577  df-termo 17892

This theorem is referenced by:  zeroopropd  49230