Theorem reldmran 50539

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 50535	. 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 7548	1 ⊢ Rel dom Ran

This proof depends on syntax axioms:  Vcvv 3450  ⦋csb 3847  ⟨cop 4590   × cxp 5653  dom cdm 5655  Rel wrel 5660  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416  1^st c1st 7985  2^nd c2nd 7986  oppCatcoppc 17800   Func cfunc 17944   FuncCat cfuc 18035   oppFunc coppf 50049   UP cup 50100   −∘_F cprcof 50300   Ran cran 50533

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 2732  ax-sep 5251  ax-pr 5398

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-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452  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 5661  df-rel 5662  df-dm 5665  df-oprab 7418  df-mpo 7419  df-ran 50535

This theorem is used by: (None)