Theorem oppcup 49704

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 17682	. . 3 ⊢ 𝐵 = (Base‘𝑂)
4		oppcup.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐸)
5		oppcup.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
6	4, 5	oppcbas 17682	. . 3 ⊢ 𝐶 = (Base‘𝑃)
7		eqid 2740	. . 3 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
8		eqid 2740	. . 3 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
9		eqid 2740	. . 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 17840	. . 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 17679	. . . 4 ⊢ (𝑊(Hom ‘𝑃)(𝐹‘𝑋)) = ((𝐹‘𝑋)𝐽𝑊)
17	14, 16	eleqtrrdi 2851	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝑃)(𝐹‘𝑋)))
18	3, 6, 7, 8, 9, 10, 12, 13, 17	isup 49677	. 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 17679	. . . . 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 17679	. . . . . 6 ⊢ (𝑋(Hom ‘𝑂)𝑦) = (𝑦𝐻𝑋)
23	22	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(Hom ‘𝑂)𝑦) = (𝑦𝐻𝑋))
24		ovtpos 8188	. . . . . . . . 9 ⊢ (𝑋tpos 𝐺𝑦) = (𝑦𝐺𝑋)
25	24	fveq1i 6835	. . . . . . . 8 ⊢ ((𝑋tpos 𝐺𝑦)‘𝑘) = ((𝑦𝐺𝑋)‘𝑘)
26	25	oveq1i 7373	. . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑊 ∈ 𝐶)
29	11	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹(𝐷 Func 𝐸)𝐺)
30	2, 5, 29	funcf1 17831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹:𝐵⟶𝐶)
31	13	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
32	30, 31	ffvelcdmd 7033	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝐶)
33		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
34	30, 33	ffvelcdmd 7033	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ 𝐶)
35	5, 27, 4, 28, 32, 34	oppcco 17681	. . . . . . 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 2787	. . . . . 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 2751	. . . . 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 3381	. . . 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 3314	. . 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 3161	. 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 280	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))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃!wreu 3343  ⟨cop 4568   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  tpos ctpos 8172  Basecbs 17177  Hom chom 17229  compcco 17230  oppCatcoppc 17675   Func cfunc 17819   UP cup 49670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-tpos 8173  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-map 8772  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-dec 12643  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-hom 17242  df-cco 17243  df-cat 17632  df-cid 17633  df-oppc 17676  df-func 17823  df-up 49671

This theorem is referenced by:  oppcup2  49705  ranup  50139  islmd  50162