Theorem cnvun

Description: The converse of a union is the union of converses. Theorem 16 of Suppes p. 62. (Contributed by NM, 25-Mar-1998) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	cnvun	$$⊢{\left(A\cup B\right)}^{-1}=A^{-1}\cup B^{-1}$$

Proof

Step	Hyp	Ref	Expression
1		df-cnv	$$⊢{\left(A\cup B\right)}^{-1}=\left\{⟨x,y⟩|y\left(A\cup B\right)x\right\}$$
2		unopab	$$⊢\left\{⟨x,y⟩|yAx\right\}\cup \left\{⟨x,y⟩|yBx\right\}=\left\{⟨x,y⟩|\left(yAx\vee yBx\right)\right\}$$
3		brun	$$⊢y\left(A\cup B\right)x\leftrightarrow \left(yAx\vee yBx\right)$$
4	3	opabbii	$$⊢\left\{⟨x,y⟩|y\left(A\cup B\right)x\right\}=\left\{⟨x,y⟩|\left(yAx\vee yBx\right)\right\}$$
5	2 4	eqtr4i	$$⊢\left\{⟨x,y⟩|yAx\right\}\cup \left\{⟨x,y⟩|yBx\right\}=\left\{⟨x,y⟩|y\left(A\cup B\right)x\right\}$$
6	1 5	eqtr4i	$$⊢{\left(A\cup B\right)}^{-1}=\left\{⟨x,y⟩|yAx\right\}\cup \left\{⟨x,y⟩|yBx\right\}$$
7		df-cnv	$$⊢A^{-1}=\left\{⟨x,y⟩|yAx\right\}$$
8		df-cnv	$$⊢B^{-1}=\left\{⟨x,y⟩|yBx\right\}$$
9	7 8	uneq12i	$$⊢A^{-1}\cup B^{-1}=\left\{⟨x,y⟩|yAx\right\}\cup \left\{⟨x,y⟩|yBx\right\}$$
10	6 9	eqtr4i	$$⊢{\left(A\cup B\right)}^{-1}=A^{-1}\cup B^{-1}$$