Theorem oduoppcciso 50266

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 2769	. 2 ⊢ (CatCat‘𝑈) = (CatCat‘𝑈)
2		eqid 2769	. 2 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2769	. 2 ⊢ (Base‘𝑂) = (Base‘𝑂)
4		eqid 2769	. 2 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2769	. 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 2769	. . . . 5 ⊢ (ODual‘𝐾) = (ODual‘𝐾)
12	11	oduprs 18358	. . . 4 ⊢ (𝐾 ∈ Proset → (ODual‘𝐾) ∈ Proset )
13	10, 12	syl 18	. . 3 ⊢ (𝜑 → (ODual‘𝐾) ∈ Proset )
14	9, 13	prstcthin 50261	. 2 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
15		prstcnid.c	. . . 4 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))
16	15, 10	prstcthin 50261	. . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
17		oduoppcbas.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
18	17	oppcthin 50138	. . 3 ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat)
19	16, 18	syl 18	. 2 ⊢ (𝜑 → 𝑂 ∈ ThinCat)
20		f1oi 6862	. . 3 ⊢ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)
21	15, 10, 9, 17	oduoppcbas 50265	. . . 4 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝑂))
22	21	f1oeq3d 6820	. . 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 236	. 2 ⊢ (𝜑 → ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝑂))
24		eqid 2769	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
25		eqid 2769	. . . . . . 7 ⊢ (le‘(ODual‘𝐾)) = (le‘(ODual‘𝐾))
26	11, 24, 25	oduleg 18348	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷)) → (𝑥(le‘(ODual‘𝐾))𝑦 ↔ 𝑦(le‘𝐾)𝑥))
27	26	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(le‘(ODual‘𝐾))𝑦 ↔ 𝑦(le‘𝐾)𝑥))
28	9	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 = (ProsetToCat‘(ODual‘𝐾)))
29	10	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐾 ∈ Proset )
30	29, 12	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (ODual‘𝐾) ∈ Proset )
31		eqidd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (le‘(ODual‘𝐾)) = (le‘(ODual‘𝐾)))
32	28, 30, 31	prstcleval 50255	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (le‘(ODual‘𝐾)) = (le‘𝐷))
33		eqidd 2770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐷) = (Hom ‘𝐷))
34		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
35		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
36	28, 30, 32, 33, 34, 35	prstchom 50262	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(le‘(ODual‘𝐾))𝑦 ↔ (𝑥(Hom ‘𝐷)𝑦) ≠ ∅))
37	15	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐶 = (ProsetToCat‘𝐾))
38		eqidd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (le‘𝐾) = (le‘𝐾))
39	37, 29, 38	prstcleval 50255	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (le‘𝐾) = (le‘𝐶))
40		eqidd 2770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐶))
41		eqid 2769	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
42	17, 41	oppcbas 17776	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝑂)
43	21, 42	eqtr4di 2822	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐶))
44	43	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Base‘𝐷) = (Base‘𝐶))
45	35, 44	eleqtrd 2871	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐶))
46	34, 44	eleqtrd 2871	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶))
47	37, 29, 39, 40, 45, 46	prstchom 50262	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑦(le‘𝐾)𝑥 ↔ (𝑦(Hom ‘𝐶)𝑥) ≠ ∅))
48	27, 36, 47	3bitr3d 312	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((𝑥(Hom ‘𝐷)𝑦) ≠ ∅ ↔ (𝑦(Hom ‘𝐶)𝑥) ≠ ∅))
49	48	necon4bid 3009	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((𝑥(Hom ‘𝐷)𝑦) = ∅ ↔ (𝑦(Hom ‘𝐶)𝑥) = ∅))
50		fvresi 7174	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝐷) → (( I ↾ (Base‘𝐷))‘𝑥) = 𝑥)
51	50	ad2antrl 740	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (( I ↾ (Base‘𝐷))‘𝑥) = 𝑥)
52		fvresi 7174	. . . . . . 7 ⊢ (𝑦 ∈ (Base‘𝐷) → (( I ↾ (Base‘𝐷))‘𝑦) = 𝑦)
53	52	ad2antll 741	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (( I ↾ (Base‘𝐷))‘𝑦) = 𝑦)
54	51, 53	oveq12d 7431	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((( I ↾ (Base‘𝐷))‘𝑥)(Hom ‘𝑂)(( I ↾ (Base‘𝐷))‘𝑦)) = (𝑥(Hom ‘𝑂)𝑦))
55		eqid 2769	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
56	55, 17	oppchom 17773	. . . . 5 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
57	54, 56	eqtrdi 2820	. . . 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 2771	. . 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 285	. 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 50154	1 ⊢ (𝜑 → 𝐷( ≃𝑐 ‘(CatCat‘𝑈))𝑂)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∅c0 4294   class class class wbr 5113   I cid 5558   ↾ cres 5666  –1-1-onto→wf1o 6538  ‘cfv 6539  (class class class)co 7413  Basecbs 17271  lecple 17319  Hom chom 17323  oppCatcoppc 17769   ≃_𝑐 ccic 17854  CatCatccatc 18157  ODualcodu 18344   Proset cproset 18350  ThinCatcthinc 50117  ProsetToCatcprstc 50249

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 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-cnex 11158  ax-resscn 11159  ax-1cn 11160  ax-icn 11161  ax-addcl 11162  ax-addrcl 11163  ax-mulcl 11164  ax-mulrcl 11165  ax-mulcom 11166  ax-addass 11167  ax-mulass 11168  ax-distr 11169  ax-i2m1 11170  ax-1ne0 11171  ax-1rid 11172  ax-rnegex 11173  ax-rrecex 11174  ax-cnre 11175  ax-pre-lttri 11176  ax-pre-lttrn 11177  ax-pre-ltadd 11178  ax-pre-mulgt0 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  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-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  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-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-supp 8159  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11445  df-neg 11446  df-nn 12236  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12865  df-fz 13538  df-struct 17209  df-sets 17226  df-slot 17244  df-ndx 17256  df-base 17272  df-ple 17332  df-hom 17336  df-cco 17337  df-cat 17726  df-cid 17727  df-oppc 17770  df-sect 17806  df-inv 17807  df-iso 17808  df-cic 17855  df-func 17917  df-idfu 17918  df-cofu 17919  df-full 17965  df-fth 17966  df-catc 18158  df-odu 18345  df-proset 18352  df-thinc 50118  df-prstc 50250

This theorem is referenced by: (None)