Theorem reldmup2 48982

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 2734	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2734	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2734	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2734	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2734	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6	1, 2, 3, 4, 5	upfval 48977	. 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 7536	1 ⊢ Rel dom (𝐷UP𝐸)

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃!wreu 3355  ⟨cop 4605  {copab 5179  dom cdm 5652  Rel wrel 5657  ‘cfv 6528  (class class class)co 7400  1^st c1st 7981  2^nd c2nd 7982  Basecbs 17215  Hom chom 17269  compcco 17270   Func cfunc 17854  UPcup 48974

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 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7983  df-2nd 7984  df-func 17858  df-up 48975

This theorem is referenced by: (None)