Theorem reldmlan 50719

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 50715	. 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 7554	1 ⊢ Rel dom Lan

This proof depends on syntax axioms:  Vcvv 3450  ⦋csb 3846  ⟨cop 4589   × cxp 5648  dom cdm 5650  Rel wrel 5655  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000   Func cfunc 18029   FuncCat cfuc 18120   UP cup 50281   −∘_F cprcof 50481   Lan clan 50713

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dm 5660  df-oprab 7424  df-mpo 7425  df-lan 50715

This theorem is used by: (None)