Theorem rtrclexlem 44877

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 7947	. . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
2		rnexg 7948	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
3	1, 2	unexd 7800	. . 3 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
4		sqxpexg 7801	. . 3 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∈ V → ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V)
5	3, 4	syl 18	. 2 ⊢ (𝑅 ∈ 𝑉 → ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V)
6		unexg 7792	. 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 700	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454   ∪ cun 4152   × cxp 5656  dom cdm 5658  ran crn 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669

This theorem is used by:  rtrclex  44878