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Theorem trclexlem 13855
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 7214 . . 3 (𝑅𝑉 → dom 𝑅 ∈ V)
2 rnexg 7215 . . 3 (𝑅𝑉 → ran 𝑅 ∈ V)
3 xpexg 7077 . . 3 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 × ran 𝑅) ∈ V)
41, 2, 3syl2anc 696 . 2 (𝑅𝑉 → (dom 𝑅 × ran 𝑅) ∈ V)
5 unexg 7076 . 2 ((𝑅𝑉 ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
64, 5mpdan 705 1 (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2103  Vcvv 3304  cun 3678   × cxp 5216  dom cdm 5218  ran crn 5219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228  df-rn 5229
This theorem is referenced by:  trclublem  13856  trclfv  13861  cnvtrcl0  38352
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