Theorem oppcup 50417

Description: The universal pair ⟨𝑋, 𝑀⟩ from a functor to an object is universal from an object to a functor in the opposite category. (Contributed by Zhi Wang, 24-Sep-2025.)

Hypotheses

Ref	Expression
oppcup.b	⊢ 𝐵 = (Base‘𝐷)
oppcup.c	⊢ 𝐶 = (Base‘𝐸)
oppcup.h	⊢ 𝐻 = (Hom ‘𝐷)
oppcup.j	⊢ 𝐽 = (Hom ‘𝐸)
oppcup.xb	⊢ ∙ = (comp‘𝐸)
oppcup.w	⊢ (𝜑 → 𝑊 ∈ 𝐶)
oppcup.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
oppcup.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
oppcup.m	⊢ (𝜑 → 𝑀 ∈ ((𝐹‘𝑋)𝐽𝑊))
oppcup.o	⊢ 𝑂 = (oppCat‘𝐷)
oppcup.p	⊢ 𝑃 = (oppCat‘𝐸)

Assertion

Ref	Expression
oppcup	⊢ (𝜑 → (𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀 ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘))))

Distinct variable groups:   𝐵,𝑔,𝑘,𝑦   𝐶,𝑔,𝑘,𝑦   𝑔,𝐹,𝑘,𝑦   𝑔,𝐺,𝑘,𝑦   𝑘,𝐻   𝑔,𝑀,𝑘,𝑦   𝑔,𝑂,𝑘,𝑦   𝑃,𝑔,𝑘,𝑦   𝑔,𝑊,𝑘,𝑦   𝑔,𝑋,𝑘,𝑦   𝜑,𝑔,𝑘,𝑦

Allowed substitution hints:   𝐷(𝑦, 𝑔, 𝑘)   ∙ (𝑦, 𝑔, 𝑘)   𝐸(𝑦, 𝑔, 𝑘)   𝐻(𝑦, 𝑔)   𝐽(𝑦, 𝑔, 𝑘)

Proof of Theorem oppcup

Step	Hyp	Ref	Expression
1		oppcup.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐷)
2		oppcup.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
3	1, 2	oppcbas 17898	. . 3 ⊢ 𝐵 = (Base‘𝑂)
4		oppcup.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐸)
5		oppcup.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
6	4, 5	oppcbas 17898	. . 3 ⊢ 𝐶 = (Base‘𝑃)
7		eqid 2760	. . 3 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
8		eqid 2760	. . 3 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
9		eqid 2760	. . 3 ⊢ (comp‘𝑃) = (comp‘𝑃)
10		oppcup.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐶)
11		oppcup.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
12	1, 4, 11	funcoppc 18056	. . 3 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)
13		oppcup.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14		oppcup.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ((𝐹‘𝑋)𝐽𝑊))
15		oppcup.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝐸)
16	15, 4	oppchom 17895	. . . 4 ⊢ (𝑊(Hom ‘𝑃)(𝐹‘𝑋)) = ((𝐹‘𝑋)𝐽𝑊)
17	14, 16	eleqtrrdi 2871	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝑃)(𝐹‘𝑋)))
18	3, 6, 7, 8, 9, 10, 12, 13, 17	isup 50390	. 2 ⊢ (𝜑 → (𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀 ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊(Hom ‘𝑃)(𝐹‘𝑦))∃!𝑘 ∈ (𝑋(Hom ‘𝑂)𝑦)𝑔 = (((𝑋tpos 𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝑃)(𝐹‘𝑦))𝑀)))
19	15, 4	oppchom 17895	. . . . 5 ⊢ (𝑊(Hom ‘𝑃)(𝐹‘𝑦)) = ((𝐹‘𝑦)𝐽𝑊)
20	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑊(Hom ‘𝑃)(𝐹‘𝑦)) = ((𝐹‘𝑦)𝐽𝑊))
21		oppcup.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐷)
22	21, 1	oppchom 17895	. . . . . 6 ⊢ (𝑋(Hom ‘𝑂)𝑦) = (𝑦𝐻𝑋)
23	22	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(Hom ‘𝑂)𝑦) = (𝑦𝐻𝑋))
24		ovtpos 8262	. . . . . . . . 9 ⊢ (𝑋tpos 𝐺𝑦) = (𝑦𝐺𝑋)
25	24	fveq1i 6887	. . . . . . . 8 ⊢ ((𝑋tpos 𝐺𝑦)‘𝑘) = ((𝑦𝐺𝑋)‘𝑘)
26	25	oveq1i 7431	. . . . . . 7 ⊢ (((𝑋tpos 𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝑃)(𝐹‘𝑦))𝑀) = (((𝑦𝐺𝑋)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝑃)(𝐹‘𝑦))𝑀)
27		oppcup.xb	. . . . . . . 8 ⊢ ∙ = (comp‘𝐸)
28	10	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑊 ∈ 𝐶)
29	11	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹(𝐷 Func 𝐸)𝐺)
30	2, 5, 29	funcf1 18047	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹:𝐵⟶𝐶)
31	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
32	30, 31	ffvelcdmd 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝐶)
33		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
34	30, 33	ffvelcdmd 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ 𝐶)
35	5, 27, 4, 28, 32, 34	oppcco 17897	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (((𝑦𝐺𝑋)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝑃)(𝐹‘𝑦))𝑀) = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘)))
36	26, 35	eqtrid 2807	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (((𝑋tpos 𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝑃)(𝐹‘𝑦))𝑀) = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘)))
37	36	eqeq2d 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑔 = (((𝑋tpos 𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝑃)(𝐹‘𝑦))𝑀) ↔ 𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘))))
38	23, 37	reueqbidv 3401	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃!𝑘 ∈ (𝑋(Hom ‘𝑂)𝑦)𝑔 = (((𝑋tpos 𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝑃)(𝐹‘𝑦))𝑀) ↔ ∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘))))
39	20, 38	raleqbidv 3334	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∀𝑔 ∈ (𝑊(Hom ‘𝑃)(𝐹‘𝑦))∃!𝑘 ∈ (𝑋(Hom ‘𝑂)𝑦)𝑔 = (((𝑋tpos 𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝑃)(𝐹‘𝑦))𝑀) ↔ ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘))))
40	39	ralbidva 3183	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊(Hom ‘𝑃)(𝐹‘𝑦))∃!𝑘 ∈ (𝑋(Hom ‘𝑂)𝑦)𝑔 = (((𝑋tpos 𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝑃)(𝐹‘𝑦))𝑀) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘))))
41	18, 40	bitrd 282	1 ⊢ (𝜑 → (𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀 ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃!wreu 3363  ⟨cop 4834   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421  tpos ctpos 8246  Basecbs 17393  Hom chom 17445  compcco 17446  oppCatcoppc 17891   Func cfunc 18035   UP cup 50383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-tpos 8247  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-er 8721  df-map 8853  df-ixp 8930  df-en 8978  df-dom 8979  df-sdom 8980  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-3 12410  df-4 12411  df-5 12412  df-6 12413  df-7 12414  df-8 12415  df-9 12416  df-n0 12611  df-z 12698  df-dec 12819  df-sets 17348  df-slot 17366  df-ndx 17378  df-base 17394  df-hom 17458  df-cco 17459  df-cat 17848  df-cid 17849  df-oppc 17892  df-func 18039  df-up 50384

This theorem is used by:  oppcup2  50418  ranup  50919  islmd  50942