Theorem reldmlan 50113

Description: The domain of Lan is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)

Assertion

Ref	Expression
reldmlan	⊢ Rel dom Lan

Proof of Theorem reldmlan

Dummy variables 𝑐 𝑑 𝑒 𝑓 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lan 50109	. 2 ⊢ Lan = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)))
2	1	reldmmpo 7493	1 ⊢ Rel dom Lan

Syntax hints:  Vcvv 3433  ⦋csb 3832  ⟨cop 4563   × cxp 5618  dom cdm 5620  Rel wrel 5625  ‘cfv 6488  (class class class)co 7359   ∈ cmpo 7361  1^st c1st 7931  2^nd c2nd 7932   Func cfunc 17816   FuncCat cfuc 17907   UP cup 49675   −∘_F cprcof 49875   Lan clan 50107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-rel 5627  df-dm 5630  df-oprab 7363  df-mpo 7364  df-lan 50109

This theorem is referenced by: (None)