Theorem relup 50392

Description: The set of universal pairs is a relation. (Contributed by Zhi Wang, 25-Sep-2025.)

Assertion

Ref	Expression
relup	⊢ Rel (𝐹(𝐷 UP 𝐸)𝑊)

Proof of Theorem relup

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

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2760	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2760	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2760	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2760	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6	1, 2, 3, 4, 5	upfval 50385	. 2 ⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ (Base‘𝐸) ↦ {⟨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	relmpoopab 8106	1 ⊢ Rel (𝐹(𝐷 UP 𝐸)𝑊)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃!wreu 3363  ⟨cop 4834  Rel wrel 5655  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17392  Hom chom 17444  compcco 17445   Func cfunc 18034   UP cup 50382

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-func 18038  df-up 50383

This theorem is used by:  up1st2nd2  50397  uobffth  50427  uobeqw  50428  isinito3  50728  rellan  50899  relran  50900  rellmd  50935  relcmd  50936  lmddu  50943  cmddu  50944