Theorem oppcup3 49208

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 2838	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
10	9	elfvexd 6852	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
11		oppcup3.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑊))
12	11	ne0d 4289	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑌)𝐽𝑊) ≠ ∅)
13		fvprc 6808	. . . . . . . . 9 ⊢ (¬ 𝐸 ∈ V → (Hom ‘𝐸) = ∅)
14	3, 13	eqtrid 2776	. . . . . . . 8 ⊢ (¬ 𝐸 ∈ V → 𝐽 = ∅)
15	14	oveqd 7357	. . . . . . 7 ⊢ (¬ 𝐸 ∈ V → ((𝐹‘𝑌)𝐽𝑊) = ((𝐹‘𝑌)∅𝑊))
16		0ov 7377	. . . . . . 7 ⊢ ((𝐹‘𝑌)∅𝑊) = ∅
17	15, 16	eqtrdi 2780	. . . . . 6 ⊢ (¬ 𝐸 ∈ V → ((𝐹‘𝑌)𝐽𝑊) = ∅)
18	17	necon1ai 2952	. . . . 5 ⊢ (((𝐹‘𝑌)𝐽𝑊) ≠ ∅ → 𝐸 ∈ V)
19	12, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
20		oppcup3.g	. . . 4 ⊢ (𝜑 → tpos 𝑇 = 𝐺)
21	7, 6, 5, 10, 19, 20	oppcuprcl2 49201	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
22	7, 20	uptpos 49197	. . 3 ⊢ (𝜑 → 𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)
23	1, 2, 3, 4, 5, 6, 21, 22	oppcup2 49207	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘)))
24	23, 8, 11	oppcup3lem 49205	1 ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑌𝐺𝑋)‘𝑘)))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃!wreu 3341  Vcvv 3433  ∅c0 4280  ⟨cop 4579   class class class wbr 5088  ‘cfv 6476  (class class class)co 7340  tpos ctpos 8149  Basecbs 17107  Hom chom 17159  compcco 17160  oppCatcoppc 17604   UP cup 49172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662  ax-cnex 11053  ax-resscn 11054  ax-1cn 11055  ax-icn 11056  ax-addcl 11057  ax-addrcl 11058  ax-mulcl 11059  ax-mulrcl 11060  ax-mulcom 11061  ax-addass 11062  ax-mulass 11063  ax-distr 11064  ax-i2m1 11065  ax-1ne0 11066  ax-1rid 11067  ax-rnegex 11068  ax-rrecex 11069  ax-cnre 11070  ax-pre-lttri 11071  ax-pre-lttrn 11072  ax-pre-ltadd 11073  ax-pre-mulgt0 11074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-tr 5196  df-id 5508  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5566  df-we 5568  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7297  df-ov 7343  df-oprab 7344  df-mpo 7345  df-om 7791  df-1st 7915  df-2nd 7916  df-tpos 8150  df-frecs 8205  df-wrecs 8236  df-recs 8285  df-rdg 8323  df-er 8616  df-map 8746  df-ixp 8816  df-en 8864  df-dom 8865  df-sdom 8866  df-pnf 11139  df-mnf 11140  df-xr 11141  df-ltxr 11142  df-le 11143  df-sub 11337  df-neg 11338  df-nn 12117  df-2 12179  df-3 12180  df-4 12181  df-5 12182  df-6 12183  df-7 12184  df-8 12185  df-9 12186  df-n0 12373  df-z 12460  df-dec 12580  df-sets 17062  df-slot 17080  df-ndx 17092  df-base 17108  df-hom 17172  df-cco 17173  df-cat 17561  df-cid 17562  df-homf 17563  df-comf 17564  df-oppc 17605  df-func 17752  df-up 49173

This theorem is referenced by: (None)