Theorem relup 48859

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

Assertion

Ref	Expression
relup	⊢ Rel (𝐹(𝐷UP𝐸)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 2737	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2737	. . 3 ⊢ (Base‘𝐸) = (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 48854	. 2 ⊢ (𝐷UP𝐸) = (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 8127	1 ⊢ Rel (𝐹(𝐷UP𝐸)𝑊)

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3061  ∃!wreu 3378  ⟨cop 4640  Rel wrel 5698  ‘cfv 6569  (class class class)co 7438  1^st c1st 8020  2^nd c2nd 8021  Basecbs 17254  Hom chom 17318  compcco 17319   Func cfunc 17914  UPcup 48851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-1st 8022  df-2nd 8023  df-func 17918  df-up 48852

This theorem is referenced by: (None)