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Theorem trclexlem 14703
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 7744 . . 3 (𝑅𝑉 → dom 𝑅 ∈ V)
2 rnexg 7745 . . 3 (𝑅𝑉 → ran 𝑅 ∈ V)
31, 2xpexd 7595 . 2 (𝑅𝑉 → (dom 𝑅 × ran 𝑅) ∈ V)
4 unexg 7593 . 2 ((𝑅𝑉 ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
53, 4mpdan 684 1 (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  Vcvv 3431  cun 3890   × cxp 5588  dom cdm 5590  ran crn 5591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5596  df-rel 5597  df-cnv 5598  df-dm 5600  df-rn 5601
This theorem is referenced by:  trclublem  14704  trclfv  14709  cnvtrcl0  41204
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