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Theorem rtrclexlem 38425
 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 7262 . . . 4 (𝑅𝑉 → dom 𝑅 ∈ V)
2 rnexg 7263 . . . 4 (𝑅𝑉 → ran 𝑅 ∈ V)
3 unexg 7124 . . . 4 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
41, 2, 3syl2anc 696 . . 3 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
5 sqxpexg 7128 . . 3 ((dom 𝑅 ∪ ran 𝑅) ∈ V → ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V)
64, 5syl 17 . 2 (𝑅𝑉 → ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V)
7 unexg 7124 . 2 ((𝑅𝑉 ∧ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V) → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)
86, 7mpdan 705 1 (𝑅𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2139  Vcvv 3340   ∪ cun 3713   × cxp 5264  dom cdm 5266  ran crn 5267 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277 This theorem is referenced by:  rtrclex  38426
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