Theorem reldmup 50227

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 50226	. 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 7546	1 ⊢ Rel dom UP

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃!wreu 3364  Vcvv 3451  ⦋csb 3847  ⟨cop 4590  {copab 5167  dom cdm 5651  Rel wrel 5656  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7988  2^nd c2nd 7989  Basecbs 17367  Hom chom 17419  compcco 17420   Func cfunc 18009   UP cup 50225

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dm 5661  df-oprab 7416  df-mpo 7417  df-up 50226

This theorem is used by:  upfval  50228