Theorem reldmran 50410

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 50406	. 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 7544	1 ⊢ Rel dom Ran

Syntax hints:  Vcvv 3455  ⦋csb 3853  ⟨cop 4595   × cxp 5659  dom cdm 5661  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981  oppCatcoppc 17762   Func cfunc 17906   FuncCat cfuc 17997   oppFunc coppf 49920   UP cup 49971   −∘_F cprcof 50171   Ran cran 50404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-dm 5671  df-oprab 7414  df-mpo 7415  df-ran 50406

This theorem is referenced by: (None)