Theorem rtrclexlem 43606

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

Proof of Theorem rtrclexlem

Step	Hyp	Ref	Expression
1		dmexg 7924	. . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
2		rnexg 7925	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
3	1, 2	unexd 7773	. . 3 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
4		sqxpexg 7774	. . 3 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∈ V → ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V)
5	3, 4	syl 17	. 2 ⊢ (𝑅 ∈ 𝑉 → ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V)
6		unexg 7762	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V) → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)
7	5, 6	mpdan 687	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3478   ∪ cun 3961   × cxp 5687  dom cdm 5689  ran crn 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700

This theorem is referenced by:  rtrclex  43607