Theorem uprcl 49681

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 2740	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		uprcl.c	. . 3 ⊢ 𝐶 = (Base‘𝐸)
3		eqid 2740	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2740	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2740	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6	1, 2, 3, 4, 5	upfval 49673	. 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 7604	1 ⊢ (𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃!wreu 3343  ⟨cop 4568  {copab 5141  ‘cfv 6492  (class class class)co 7363  1^st c1st 7936  2^nd c2nd 7937  Basecbs 17177  Hom chom 17229  compcco 17230   Func cfunc 17819   UP cup 49670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  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-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-func 17823  df-up 49671

This theorem is referenced by:  up1st2nd  49682  uprcl2  49686  uprcl3  49687  uprcl2a  49700  lanval2  50124  ranval2  50127  ranval3  50128  lmdfval2  50152  cmdfval2  50153