Theorem reldmran 49867

Description: The domain of Ran is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)

Assertion

Ref	Expression
reldmran	⊢ Rel dom Ran

Proof of Theorem reldmran

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

Step	Hyp	Ref	Expression
1		df-ran 49863	. 2 ⊢ Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))
2	1	reldmmpo 7492	1 ⊢ Rel dom Ran

Syntax hints:  Vcvv 3440  ⦋csb 3849  ⟨cop 4586   × cxp 5622  dom cdm 5624  Rel wrel 5629  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932  oppCatcoppc 17634   Func cfunc 17778   FuncCat cfuc 17869   oppFunc coppf 49377   UP cup 49428   −∘_F cprcof 49628   Ran cran 49861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634  df-oprab 7362  df-mpo 7363  df-ran 49863

This theorem is referenced by: (None)