Theorem reldmcmd 50970

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 50968	. 2 ⊢ Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
2	1	reldmmpo 7561	1 ⊢ Rel dom Colimit

This proof depends on syntax axioms:  Vcvv 3453   ↦ cmpt 5188  dom cdm 5654  Rel wrel 5659  (class class class)co 7427   Func cfunc 18041   FuncCat cfuc 18132  Δ_funccdiag 18398   UP cup 50428   Colimit ccmd 50966

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-pr 5394

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 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-dm 5664  df-oprab 7431  df-mpo 7432  df-cmd 50968

This theorem is used by:  cmdfval  50972