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Theorem rtrclexlem 42948
Description: Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.)
Assertion
Ref Expression
rtrclexlem (𝑅𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)

Proof of Theorem rtrclexlem
StepHypRef Expression
1 dmexg 7891 . . . 4 (𝑅𝑉 → dom 𝑅 ∈ V)
2 rnexg 7892 . . . 4 (𝑅𝑉 → ran 𝑅 ∈ V)
31, 2unexd 7738 . . 3 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
4 sqxpexg 7739 . . 3 ((dom 𝑅 ∪ ran 𝑅) ∈ V → ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V)
53, 4syl 17 . 2 (𝑅𝑉 → ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V)
6 unexg 7733 . 2 ((𝑅𝑉 ∧ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V) → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)
75, 6mpdan 684 1 (𝑅𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3468  cun 3941   × cxp 5667  dom cdm 5669  ran crn 5670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680
This theorem is referenced by:  rtrclex  42949
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