Theorem rtrclexlem 44152

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 7876	. . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
2		rnexg 7877	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
3	1, 2	unexd 7731	. . 3 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
4		sqxpexg 7732	. . 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 7720	. 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 697	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 3453   ∪ cun 3900   × cxp 5641  dom cdm 5643  ran crn 5644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-xp 5649  df-rel 5650  df-cnv 5651  df-dm 5653  df-rn 5654

This theorem is referenced by:  rtrclex  44153