Theorem uprcl 49671

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 2737	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		uprcl.c	. . 3 ⊢ 𝐶 = (Base‘𝐸)
3		eqid 2737	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2737	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2737	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6	1, 2, 3, 4, 5	upfval 49663	. 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 7601	1 ⊢ (𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3341  ⟨cop 4574  {copab 5148  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17170  Hom chom 17222  compcco 17223   Func cfunc 17812   UP cup 49660

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3053  df-rex 3063  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-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-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-func 17816  df-up 49661

This theorem is referenced by:  up1st2nd  49672  uprcl2  49676  uprcl3  49677  uprcl2a  49690  lanval2  50114  ranval2  50117  ranval3  50118  lmdfval2  50142  cmdfval2  50143