Theorem uprcl 49179

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

Hypothesis

Ref	Expression
uprcl.c	⊢ 𝐶 = (Base‘𝐸)

Assertion

Ref	Expression
uprcl	⊢ (𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ 𝐶))

Proof of Theorem uprcl

Dummy variables 𝑓 𝑔 𝑘 𝑚 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		uprcl.c	. . 3 ⊢ 𝐶 = (Base‘𝐸)
3		eqid 2729	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2729	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2729	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6	1, 2, 3, 4, 5	upfval 49171	. 2 ⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ (Base‘𝐷) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐸)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐷)∀𝑔 ∈ (𝑤(Hom ‘𝐸)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐷)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐸)((1st ‘𝑓)‘𝑦))𝑚))})
7	6	elmpocl 7590	1 ⊢ (𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃!wreu 3341  ⟨cop 4583  {copab 5154  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  Basecbs 17120  Hom chom 17172  compcco 17173   Func cfunc 17761   UP cup 49168

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  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 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-func 17765  df-up 49169

This theorem is referenced by:  up1st2nd  49180  uprcl2  49184  uprcl3  49185  uprcl2a  49198  lanval2  49622  ranval2  49625  ranval3  49626  lmdfval2  49650  cmdfval2  49651