Theorem trclexlem

Description: Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 5-May-2020)

	Ref	Expression
Assertion	trclexlem	$$⊢R\in V\to R\cup \left(\mathrm{dom}R\times \mathrm{ran}R\right)\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		dmexg	$$⊢R\in V\to \mathrm{dom}R\in \mathrm{V}$$
2		rnexg	$$⊢R\in V\to \mathrm{ran}R\in \mathrm{V}$$
3	1 2	xpexd	$$⊢R\in V\to \mathrm{dom}R\times \mathrm{ran}R\in \mathrm{V}$$
4		unexg	$$⊢\left(R\in V\wedge \mathrm{dom}R\times \mathrm{ran}R\in \mathrm{V}\right)\to R\cup \left(\mathrm{dom}R\times \mathrm{ran}R\right)\in \mathrm{V}$$
5	3 4	mpdan	$$⊢R\in V\to R\cup \left(\mathrm{dom}R\times \mathrm{ran}R\right)\in \mathrm{V}$$