Theorem oppcup 50259

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 17872	. . 3 ⊢ 𝐵 = (Base‘𝑂)
4		oppcup.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐸)
5		oppcup.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
6	4, 5	oppcbas 17872	. . 3 ⊢ 𝐶 = (Base‘𝑃)
7		eqid 2761	. . 3 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
8		eqid 2761	. . 3 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
9		eqid 2761	. . 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 18030	. . 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 17869	. . . 4 ⊢ (𝑊(Hom ‘𝑃)(𝐹‘𝑋)) = ((𝐹‘𝑋)𝐽𝑊)
17	14, 16	eleqtrrdi 2872	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝑃)(𝐹‘𝑋)))
18	3, 6, 7, 8, 9, 10, 12, 13, 17	isup 50232	. 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 17869	. . . . 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 17869	. . . . . 6 ⊢ (𝑋(Hom ‘𝑂)𝑦) = (𝑦𝐻𝑋)
23	22	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(Hom ‘𝑂)𝑦) = (𝑦𝐻𝑋))
24		ovtpos 8242	. . . . . . . . 9 ⊢ (𝑋tpos 𝐺𝑦) = (𝑦𝐺𝑋)
25	24	fveq1i 6878	. . . . . . . 8 ⊢ ((𝑋tpos 𝐺𝑦)‘𝑘) = ((𝑦𝐺𝑋)‘𝑘)
26	25	oveq1i 7422	. . . . . . 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 18021	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹:𝐵⟶𝐶)
31	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
32	30, 31	ffvelcdmd 7077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝐶)
33		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
34	30, 33	ffvelcdmd 7077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ 𝐶)
35	5, 27, 4, 28, 32, 34	oppcco 17871	. . . . . . 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 2808	. . . . . 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 2772	. . . . 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 3402	. . . 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 3335	. . 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 3184	. 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 1570   ∈ wcel 2145  ∀wral 3077  ∃!wreu 3364  ⟨cop 4590   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  tpos ctpos 8226  Basecbs 17367  Hom chom 17419  compcco 17420  oppCatcoppc 17865   Func cfunc 18009   UP cup 50225

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-tpos 8227  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-map 8833  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-z 12675  df-dec 12796  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-hom 17432  df-cco 17433  df-cat 17822  df-cid 17823  df-oppc 17866  df-func 18013  df-up 50226

This theorem is used by:  oppcup2  50260  ranup  50694  islmd  50717