Theorem trclexlem 14931

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 7855	. . 3 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
2		rnexg 7856	. . 3 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
3	1, 2	xpexd 7708	. 2 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 × ran 𝑅) ∈ V)
4		unexg 7700	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
5	3, 4	mpdan 688	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   × cxp 5632  dom cdm 5634  ran crn 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645

This theorem is referenced by:  trclublem  14932  trclfv  14937  cnvtrcl0  44011