Theorem reldmup 49083

Description: The domain of UP is a relation. (Contributed by Zhi Wang, 25-Sep-2025.)

Assertion

Ref	Expression
reldmup	⊢ Rel dom UP

Proof of Theorem reldmup

Dummy variables 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑗 𝑘 𝑚 𝑜 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-up 49082	. 2 ⊢ UP = (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}))
2	1	reldmmpo 7530	1 ⊢ Rel dom UP

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3046  ∃!wreu 3355  Vcvv 3455  ⦋csb 3870  ⟨cop 4603  {copab 5177  dom cdm 5646  Rel wrel 5651  ‘cfv 6519  (class class class)co 7394   ∈ cmpo 7396  1^st c1st 7975  2^nd c2nd 7976  Basecbs 17185  Hom chom 17237  compcco 17238   Func cfunc 17822   UP cup 49081

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 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-br 5116  df-opab 5178  df-xp 5652  df-rel 5653  df-dm 5656  df-oprab 7398  df-mpo 7399  df-up 49082

This theorem is referenced by:  upfval  49084