Theorem reldmcmd 50755

Description: The domain of Colimit is a relation. (Contributed by Zhi Wang, 12-Nov-2025.)

Assertion

Ref	Expression
reldmcmd	⊢ Rel dom Colimit

Proof of Theorem reldmcmd

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

Step	Hyp	Ref	Expression
1		df-cmd 50753	. 2 ⊢ Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
2	1	reldmmpo 7554	1 ⊢ Rel dom Colimit

This proof depends on syntax axioms:  Vcvv 3451   ↦ cmpt 5186  dom cdm 5651  Rel wrel 5656  (class class class)co 7420   Func cfunc 18029   FuncCat cfuc 18120  Δ_funccdiag 18386   UP cup 50280   Colimit ccmd 50751

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dm 5661  df-oprab 7424  df-mpo 7425  df-cmd 50753

This theorem is used by:  cmdfval  50757