Theorem uprcl5 49317

Description: Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.)

Hypotheses

Ref	Expression
uprcl2.x	⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
uprcl5.j	⊢ 𝐽 = (Hom ‘𝐸)

Assertion

Ref	Expression
uprcl5	⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋)))

Proof of Theorem uprcl5

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

Step	Hyp	Ref	Expression
1		uprcl2.x	. . 3 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
2		eqid 2733	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2733	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		eqid 2733	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		uprcl5.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
6		eqid 2733	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
7	1, 3	uprcl3 49315	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
8	1	uprcl2 49314	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
9	2, 3, 4, 5, 6, 7, 8	isuplem 49304	. . 3 ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀 ↔ ((𝑋 ∈ (Base‘𝐷) ∧ 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋))) ∧ ∀𝑦 ∈ (Base‘𝐷)∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋(Hom ‘𝐷)𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑦))𝑀))))
10	1, 9	mpbid 232	. 2 ⊢ (𝜑 → ((𝑋 ∈ (Base‘𝐷) ∧ 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋))) ∧ ∀𝑦 ∈ (Base‘𝐷)∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋(Hom ‘𝐷)𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑦))𝑀)))
11	10	simplrd 769	1 ⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃!wreu 3345  ⟨cop 4581   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  Hom chom 17174  compcco 17175   UP cup 49298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-func 17767  df-up 49299

This theorem is referenced by:  isup2  49319  upeu3  49320  upeu4  49321  oppcuprcl5  49326  uptri  49339  uptrar  49341  uptrai  49342  uptr2  49346  isinito3  49625  lanrcl5  49760  iscmd  49791  lmddu  49792