Theorem reldmup2 48911

Description: The domain of (𝐷UP𝐸) is a relation. (Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
reldmup2	⊢ Rel dom (𝐷UP𝐸)

Proof of Theorem reldmup2

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

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2736	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2736	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2736	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2736	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6	1, 2, 3, 4, 5	upfval 48906	. 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	reldmmpo 7564	1 ⊢ Rel dom (𝐷UP𝐸)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3060  ∃!wreu 3377  ⟨cop 4630  {copab 5203  dom cdm 5683  Rel wrel 5688  ‘cfv 6559  (class class class)co 7429  1^st c1st 8008  2^nd c2nd 8009  Basecbs 17243  Hom chom 17304  compcco 17305   Func cfunc 17895  UPcup 48903

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-func 17899  df-up 48904

This theorem is referenced by: (None)