Theorem relup 50117

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 2762	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2762	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2762	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2762	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2762	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6	1, 2, 3, 4, 5	upfval 50110	. 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 8095	1 ⊢ Rel (𝐹(𝐷 UP 𝐸)𝑊)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃!wreu 3365  ⟨cop 4593  Rel wrel 5664  ‘cfv 6537  (class class class)co 7417  1^st c1st 7988  2^nd c2nd 7989  Basecbs 17307  Hom chom 17359  compcco 17360   Func cfunc 17949   UP cup 50107

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-func 17953  df-up 50108

This theorem is used by:  up1st2nd2  50122  uobffth  50152  uobeqw  50153  isinito3  50434  rellan  50557  relran  50558  rellmd  50593  relcmd  50594  lmddu  50601  cmddu  50602