Theorem oduoppcciso 49666

Description: The dual of a preordered set and the opposite category are category-isomorphic. Example 3.6(1) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-Sep-2025.)

Hypotheses

Ref	Expression
prstcnid.c	⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))
prstcnid.k	⊢ (𝜑 → 𝐾 ∈ Proset )
oduoppcbas.d	⊢ (𝜑 → 𝐷 = (ProsetToCat‘(ODual‘𝐾)))
oduoppcbas.o	⊢ 𝑂 = (oppCat‘𝐶)
oduoppcciso.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
oduoppcciso.d	⊢ (𝜑 → 𝐷 ∈ 𝑈)
oduoppcciso.o	⊢ (𝜑 → 𝑂 ∈ 𝑈)

Assertion

Ref	Expression
oduoppcciso	⊢ (𝜑 → 𝐷( ≃𝑐 ‘(CatCat‘𝑈))𝑂)

Proof of Theorem oduoppcciso

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. 2 ⊢ (CatCat‘𝑈) = (CatCat‘𝑈)
2		eqid 2731	. 2 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2731	. 2 ⊢ (Base‘𝑂) = (Base‘𝑂)
4		eqid 2731	. 2 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2731	. 2 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
6		oduoppcciso.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
7		oduoppcciso.d	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
8		oduoppcciso.o	. 2 ⊢ (𝜑 → 𝑂 ∈ 𝑈)
9		oduoppcbas.d	. . 3 ⊢ (𝜑 → 𝐷 = (ProsetToCat‘(ODual‘𝐾)))
10		prstcnid.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Proset )
11		eqid 2731	. . . . 5 ⊢ (ODual‘𝐾) = (ODual‘𝐾)
12	11	oduprs 18206	. . . 4 ⊢ (𝐾 ∈ Proset → (ODual‘𝐾) ∈ Proset )
13	10, 12	syl 17	. . 3 ⊢ (𝜑 → (ODual‘𝐾) ∈ Proset )
14	9, 13	prstcthin 49661	. 2 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
15		prstcnid.c	. . . 4 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))
16	15, 10	prstcthin 49661	. . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
17		oduoppcbas.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
18	17	oppcthin 49538	. . 3 ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat)
19	16, 18	syl 17	. 2 ⊢ (𝜑 → 𝑂 ∈ ThinCat)
20		f1oi 6801	. . 3 ⊢ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)
21	15, 10, 9, 17	oduoppcbas 49665	. . . 4 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝑂))
22	21	f1oeq3d 6760	. . 3 ⊢ (𝜑 → (( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷) ↔ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝑂)))
23	20, 22	mpbii 233	. 2 ⊢ (𝜑 → ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝑂))
24		eqid 2731	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
25		eqid 2731	. . . . . . 7 ⊢ (le‘(ODual‘𝐾)) = (le‘(ODual‘𝐾))
26	11, 24, 25	oduleg 18196	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷)) → (𝑥(le‘(ODual‘𝐾))𝑦 ↔ 𝑦(le‘𝐾)𝑥))
27	26	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(le‘(ODual‘𝐾))𝑦 ↔ 𝑦(le‘𝐾)𝑥))
28	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 = (ProsetToCat‘(ODual‘𝐾)))
29	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐾 ∈ Proset )
30	29, 12	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (ODual‘𝐾) ∈ Proset )
31		eqidd 2732	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (le‘(ODual‘𝐾)) = (le‘(ODual‘𝐾)))
32	28, 30, 31	prstcleval 49655	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (le‘(ODual‘𝐾)) = (le‘𝐷))
33		eqidd 2732	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐷) = (Hom ‘𝐷))
34		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
35		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
36	28, 30, 32, 33, 34, 35	prstchom 49662	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(le‘(ODual‘𝐾))𝑦 ↔ (𝑥(Hom ‘𝐷)𝑦) ≠ ∅))
37	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐶 = (ProsetToCat‘𝐾))
38		eqidd 2732	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (le‘𝐾) = (le‘𝐾))
39	37, 29, 38	prstcleval 49655	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (le‘𝐾) = (le‘𝐶))
40		eqidd 2732	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐶))
41		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
42	17, 41	oppcbas 17624	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝑂)
43	21, 42	eqtr4di 2784	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐶))
44	43	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Base‘𝐷) = (Base‘𝐶))
45	35, 44	eleqtrd 2833	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐶))
46	34, 44	eleqtrd 2833	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶))
47	37, 29, 39, 40, 45, 46	prstchom 49662	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑦(le‘𝐾)𝑥 ↔ (𝑦(Hom ‘𝐶)𝑥) ≠ ∅))
48	27, 36, 47	3bitr3d 309	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((𝑥(Hom ‘𝐷)𝑦) ≠ ∅ ↔ (𝑦(Hom ‘𝐶)𝑥) ≠ ∅))
49	48	necon4bid 2973	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((𝑥(Hom ‘𝐷)𝑦) = ∅ ↔ (𝑦(Hom ‘𝐶)𝑥) = ∅))
50		fvresi 7107	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝐷) → (( I ↾ (Base‘𝐷))‘𝑥) = 𝑥)
51	50	ad2antrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (( I ↾ (Base‘𝐷))‘𝑥) = 𝑥)
52		fvresi 7107	. . . . . . 7 ⊢ (𝑦 ∈ (Base‘𝐷) → (( I ↾ (Base‘𝐷))‘𝑦) = 𝑦)
53	52	ad2antll 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (( I ↾ (Base‘𝐷))‘𝑦) = 𝑦)
54	51, 53	oveq12d 7364	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((( I ↾ (Base‘𝐷))‘𝑥)(Hom ‘𝑂)(( I ↾ (Base‘𝐷))‘𝑦)) = (𝑥(Hom ‘𝑂)𝑦))
55		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
56	55, 17	oppchom 17621	. . . . 5 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
57	54, 56	eqtrdi 2782	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((( I ↾ (Base‘𝐷))‘𝑥)(Hom ‘𝑂)(( I ↾ (Base‘𝐷))‘𝑦)) = (𝑦(Hom ‘𝐶)𝑥))
58	57	eqeq1d 2733	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (((( I ↾ (Base‘𝐷))‘𝑥)(Hom ‘𝑂)(( I ↾ (Base‘𝐷))‘𝑦)) = ∅ ↔ (𝑦(Hom ‘𝐶)𝑥) = ∅))
59	49, 58	bitr4d 282	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((𝑥(Hom ‘𝐷)𝑦) = ∅ ↔ ((( I ↾ (Base‘𝐷))‘𝑥)(Hom ‘𝑂)(( I ↾ (Base‘𝐷))‘𝑦)) = ∅))
60	1, 2, 3, 4, 5, 6, 7, 8, 14, 19, 23, 59	thinccisod 49554	1 ⊢ (𝜑 → 𝐷( ≃𝑐 ‘(CatCat‘𝑈))𝑂)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∅c0 4280   class class class wbr 5089   I cid 5508   ↾ cres 5616  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  lecple 17168  Hom chom 17172  oppCatcoppc 17617   ≃_𝑐 ccic 17702  CatCatccatc 18005  ODualcodu 18192   Proset cproset 18198  ThinCatcthinc 49517  ProsetToCatcprstc 49649

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 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  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-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-tpos 8156  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ple 17181  df-hom 17185  df-cco 17186  df-cat 17574  df-cid 17575  df-oppc 17618  df-sect 17654  df-inv 17655  df-iso 17656  df-cic 17703  df-func 17765  df-idfu 17766  df-cofu 17767  df-full 17813  df-fth 17814  df-catc 18006  df-odu 18193  df-proset 18200  df-thinc 49518  df-prstc 49650

This theorem is referenced by: (None)