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Theorem trclexlem 14112
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 7358 . . 3 (𝑅𝑉 → dom 𝑅 ∈ V)
2 rnexg 7359 . . 3 (𝑅𝑉 → ran 𝑅 ∈ V)
31, 2xpexd 7221 . 2 (𝑅𝑉 → (dom 𝑅 × ran 𝑅) ∈ V)
4 unexg 7219 . 2 ((𝑅𝑉 ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
53, 4mpdan 680 1 (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166  Vcvv 3414  cun 3796   × cxp 5340  dom cdm 5342  ran crn 5343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-dm 5352  df-rn 5353
This theorem is referenced by:  trclublem  14113  trclfv  14118  cnvtrcl0  38774
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