Theorem oppcup3 49872

Description: The universal property for the universal pair ⟨𝑋, 𝑀⟩ from a functor to an object, expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.)

Hypotheses

Ref	Expression
oppcup3.b	⊢ 𝐵 = (Base‘𝐷)
oppcup3.h	⊢ 𝐻 = (Hom ‘𝐷)
oppcup3.j	⊢ 𝐽 = (Hom ‘𝐸)
oppcup3.xb	⊢ ∙ = (comp‘𝐸)
oppcup3.o	⊢ 𝑂 = (oppCat‘𝐷)
oppcup3.p	⊢ 𝑃 = (oppCat‘𝐸)
oppcup3.x	⊢ (𝜑 → 𝑋(⟨𝐹, 𝑇⟩(𝑂 UP 𝑃)𝑊)𝑀)
oppcup3.g	⊢ (𝜑 → tpos 𝑇 = 𝐺)
oppcup3.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
oppcup3.n	⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑊))

Assertion

Ref	Expression
oppcup3	⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑌𝐺𝑋)‘𝑘)))

Distinct variable groups:   ∙ ,𝑘   𝐵,𝑘   𝑘,𝐸   𝑘,𝐹   𝑘,𝐺   𝑘,𝐻   𝑘,𝑀   𝑘,𝑁   𝑘,𝑂   𝑃,𝑘   𝑘,𝑊   𝑘,𝑋   𝑘,𝑌   𝜑,𝑘

Allowed substitution hints:   𝐷(𝑘)   𝑇(𝑘)   𝐽(𝑘)

Proof of Theorem oppcup3

Dummy variables 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppcup3.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
2		oppcup3.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
3		oppcup3.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐸)
4		oppcup3.xb	. . 3 ⊢ ∙ = (comp‘𝐸)
5		oppcup3.o	. . 3 ⊢ 𝑂 = (oppCat‘𝐷)
6		oppcup3.p	. . 3 ⊢ 𝑃 = (oppCat‘𝐸)
7		oppcup3.x	. . . 4 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝑇⟩(𝑂 UP 𝑃)𝑊)𝑀)
8		oppcup3.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	8, 1	eleqtrdi 2879	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
10	9	elfvexd 6918	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
11		oppcup3.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑊))
12	11	ne0d 4303	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑌)𝐽𝑊) ≠ ∅)
13		fvprc 6874	. . . . . . . . 9 ⊢ (¬ 𝐸 ∈ V → (Hom ‘𝐸) = ∅)
14	3, 13	eqtrid 2816	. . . . . . . 8 ⊢ (¬ 𝐸 ∈ V → 𝐽 = ∅)
15	14	oveqd 7428	. . . . . . 7 ⊢ (¬ 𝐸 ∈ V → ((𝐹‘𝑌)𝐽𝑊) = ((𝐹‘𝑌)∅𝑊))
16		0ov 7448	. . . . . . 7 ⊢ ((𝐹‘𝑌)∅𝑊) = ∅
17	15, 16	eqtrdi 2820	. . . . . 6 ⊢ (¬ 𝐸 ∈ V → ((𝐹‘𝑌)𝐽𝑊) = ∅)
18	17	necon1ai 2991	. . . . 5 ⊢ (((𝐹‘𝑌)𝐽𝑊) ≠ ∅ → 𝐸 ∈ V)
19	12, 18	syl 18	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
20		oppcup3.g	. . . 4 ⊢ (𝜑 → tpos 𝑇 = 𝐺)
21	7, 6, 5, 10, 19, 20	oppcuprcl2 49865	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
22	7, 20	uptpos 49861	. . 3 ⊢ (𝜑 → 𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)
23	1, 2, 3, 4, 5, 6, 21, 22	oppcup2 49871	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘)))
24	23, 8, 11	oppcup3lem 49869	1 ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑌𝐺𝑋)‘𝑘)))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃!wreu 3374  Vcvv 3463  ∅c0 4294  ⟨cop 4600   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  tpos ctpos 8221  Basecbs 17269  Hom chom 17321  compcco 17322  oppCatcoppc 17767   UP cup 49836

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 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

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-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-map 8826  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-hom 17334  df-cco 17335  df-cat 17724  df-cid 17725  df-homf 17726  df-comf 17727  df-oppc 17768  df-func 17915  df-up 49837

This theorem is referenced by: (None)