MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trclexlem Structured version   Visualization version   GIF version

Theorem trclexlem 14353
Description: Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 5-May-2020.)
Assertion
Ref Expression
trclexlem (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)

Proof of Theorem trclexlem
StepHypRef Expression
1 dmexg 7612 . . 3 (𝑅𝑉 → dom 𝑅 ∈ V)
2 rnexg 7613 . . 3 (𝑅𝑉 → ran 𝑅 ∈ V)
31, 2xpexd 7473 . 2 (𝑅𝑉 → (dom 𝑅 × ran 𝑅) ∈ V)
4 unexg 7471 . 2 ((𝑅𝑉 ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
53, 4mpdan 685 1 (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  Vcvv 3494  cun 3933   × cxp 5552  dom cdm 5554  ran crn 5555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565
This theorem is referenced by:  trclublem  14354  trclfv  14359  cnvtrcl0  39986
  Copyright terms: Public domain W3C validator