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	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)

Proof of Theorem trclexlem

Step	Hyp	Ref	Expression
1		dmexg 7744	. . 3 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
2		rnexg 7745	. . 3 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
3	1, 2	xpexd 7595	. 2 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 × ran 𝑅) ∈ V)
4		unexg 7593	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
5	3, 4	mpdan 684	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)

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