Theorem brab

Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999)

	Ref	Expression
Hypotheses	opelopab.1	$$⊢A\in \mathrm{V}$$
	opelopab.2	$$⊢B\in \mathrm{V}$$
	opelopab.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	opelopab.4	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
	brab.5	$$⊢R=\left\{⟨x,y⟩|\phi \right\}$$
Assertion	brab	$$⊢ARB\leftrightarrow \chi$$

Proof

Step	Hyp	Ref	Expression
1		opelopab.1	$$⊢A\in \mathrm{V}$$
2		opelopab.2	$$⊢B\in \mathrm{V}$$
3		opelopab.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
4		opelopab.4	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
5		brab.5	$$⊢R=\left\{⟨x,y⟩|\phi \right\}$$
6	3 4 5	brabg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(ARB\leftrightarrow \chi \right)$$
7	1 2 6	mp2an	$$⊢ARB\leftrightarrow \chi$$