Theorem oppcup3 49188

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 2839	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
10	9	elfvexd 6899	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
11		oppcup3.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑊))
12	11	ne0d 4307	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑌)𝐽𝑊) ≠ ∅)
13		fvprc 6852	. . . . . . . . 9 ⊢ (¬ 𝐸 ∈ V → (Hom ‘𝐸) = ∅)
14	3, 13	eqtrid 2777	. . . . . . . 8 ⊢ (¬ 𝐸 ∈ V → 𝐽 = ∅)
15	14	oveqd 7406	. . . . . . 7 ⊢ (¬ 𝐸 ∈ V → ((𝐹‘𝑌)𝐽𝑊) = ((𝐹‘𝑌)∅𝑊))
16		0ov 7426	. . . . . . 7 ⊢ ((𝐹‘𝑌)∅𝑊) = ∅
17	15, 16	eqtrdi 2781	. . . . . 6 ⊢ (¬ 𝐸 ∈ V → ((𝐹‘𝑌)𝐽𝑊) = ∅)
18	17	necon1ai 2953	. . . . 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 49181	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
22	7, 20	uptpos 49177	. . 3 ⊢ (𝜑 → 𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)
23	1, 2, 3, 4, 5, 6, 21, 22	oppcup2 49187	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘)))
24	23, 8, 11	oppcup3lem 49185	1 ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑌𝐺𝑋)‘𝑘)))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃!wreu 3354  Vcvv 3450  ∅c0 4298  ⟨cop 4597   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  tpos ctpos 8206  Basecbs 17185  Hom chom 17237  compcco 17238  oppCatcoppc 17678   UP cup 49152

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-er 8673  df-map 8803  df-ixp 8873  df-en 8921  df-dom 8922  df-sdom 8923  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-homf 17637  df-comf 17638  df-oppc 17679  df-func 17826  df-up 49153

This theorem is referenced by: (None)