Theorem oppcup3 49699

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 2847	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
10	9	elfvexd 6871	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
11		oppcup3.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑊))
12	11	ne0d 4283	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑌)𝐽𝑊) ≠ ∅)
13		fvprc 6827	. . . . . . . . 9 ⊢ (¬ 𝐸 ∈ V → (Hom ‘𝐸) = ∅)
14	3, 13	eqtrid 2784	. . . . . . . 8 ⊢ (¬ 𝐸 ∈ V → 𝐽 = ∅)
15	14	oveqd 7378	. . . . . . 7 ⊢ (¬ 𝐸 ∈ V → ((𝐹‘𝑌)𝐽𝑊) = ((𝐹‘𝑌)∅𝑊))
16		0ov 7398	. . . . . . 7 ⊢ ((𝐹‘𝑌)∅𝑊) = ∅
17	15, 16	eqtrdi 2788	. . . . . 6 ⊢ (¬ 𝐸 ∈ V → ((𝐹‘𝑌)𝐽𝑊) = ∅)
18	17	necon1ai 2960	. . . . 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 49692	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
22	7, 20	uptpos 49688	. . 3 ⊢ (𝜑 → 𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)
23	1, 2, 3, 4, 5, 6, 21, 22	oppcup2 49698	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘)))
24	23, 8, 11	oppcup3lem 49696	1 ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑌𝐺𝑋)‘𝑘)))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃!wreu 3341  Vcvv 3430  ∅c0 4274  ⟨cop 4574   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  tpos ctpos 8169  Basecbs 17173  Hom chom 17225  compcco 17226  oppCatcoppc 17671   UP cup 49663

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-hom 17238  df-cco 17239  df-cat 17628  df-cid 17629  df-homf 17630  df-comf 17631  df-oppc 17672  df-func 17819  df-up 49664

This theorem is referenced by: (None)