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Theorem reldmran 50186
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.
StepHypRef Expression
1 df-ran 50182 . 2 Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ (1st𝑝) / 𝑐(2nd𝑝) / 𝑑(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))
21reldmmpo 7524 1 Rel dom Ran
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3453  csb 3852  cop 4587   × cxp 5643  dom cdm 5645  Rel wrel 5650  cfv 6515  (class class class)co 7390  cmpo 7392  1st c1st 7962  2nd c2nd 7963  oppCatcoppc 17724   Func cfunc 17868   FuncCat cfuc 17959   oppFunc coppf 49696   UP cup 49747   −∘F cprcof 49947   Ran cran 50180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-dm 5655  df-oprab 7394  df-mpo 7395  df-ran 50182
This theorem is referenced by: (None)
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