Theorem oppcup 49682

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 17684	. . 3 ⊢ 𝐵 = (Base‘𝑂)
4		oppcup.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐸)
5		oppcup.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
6	4, 5	oppcbas 17684	. . 3 ⊢ 𝐶 = (Base‘𝑃)
7		eqid 2736	. . 3 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
8		eqid 2736	. . 3 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
9		eqid 2736	. . 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 17842	. . 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 17681	. . . 4 ⊢ (𝑊(Hom ‘𝑃)(𝐹‘𝑋)) = ((𝐹‘𝑋)𝐽𝑊)
17	14, 16	eleqtrrdi 2847	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝑃)(𝐹‘𝑋)))
18	3, 6, 7, 8, 9, 10, 12, 13, 17	isup 49655	. 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 17681	. . . . 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 17681	. . . . . 6 ⊢ (𝑋(Hom ‘𝑂)𝑦) = (𝑦𝐻𝑋)
23	22	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(Hom ‘𝑂)𝑦) = (𝑦𝐻𝑋))
24		ovtpos 8191	. . . . . . . . 9 ⊢ (𝑋tpos 𝐺𝑦) = (𝑦𝐺𝑋)
25	24	fveq1i 6841	. . . . . . . 8 ⊢ ((𝑋tpos 𝐺𝑦)‘𝑘) = ((𝑦𝐺𝑋)‘𝑘)
26	25	oveq1i 7377	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑊 ∈ 𝐶)
29	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹(𝐷 Func 𝐸)𝐺)
30	2, 5, 29	funcf1 17833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹:𝐵⟶𝐶)
31	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
32	30, 31	ffvelcdmd 7037	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝐶)
33		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
34	30, 33	ffvelcdmd 7037	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ 𝐶)
35	5, 27, 4, 28, 32, 34	oppcco 17683	. . . . . . 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 2783	. . . . . 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 2747	. . . . 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 3378	. . . 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 3311	. . 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 3158	. 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 279	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃!wreu 3340  ⟨cop 4573   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  tpos ctpos 8175  Basecbs 17179  Hom chom 17231  compcco 17232  oppCatcoppc 17677   Func cfunc 17821   UP cup 49648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17634  df-cid 17635  df-oppc 17678  df-func 17825  df-up 49649

This theorem is referenced by:  oppcup2  49683  ranup  50117  islmd  50140