Theorem dom2

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. C and D can be read C ( x ) and D ( y ) , as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003)

	Ref	Expression
Hypotheses	dom2.1	$$⊢x\in A\to C\in B$$
	dom2.2	$$⊢\left(x\in A\wedge y\in A\right)\to \left(C=D\leftrightarrow x=y\right)$$
Assertion	dom2	$$⊢B\in V\to A\preccurlyeq B$$

Proof

Step	Hyp	Ref	Expression
1		dom2.1	$$⊢x\in A\to C\in B$$
2		dom2.2	$$⊢\left(x\in A\wedge y\in A\right)\to \left(C=D\leftrightarrow x=y\right)$$
3		eqid	$$⊢A=A$$
4	1	a1i	$$⊢A=A\to \left(x\in A\to C\in B\right)$$
5	2	a1i	$$⊢A=A\to \left(\left(x\in A\wedge y\in A\right)\to \left(C=D\leftrightarrow x=y\right)\right)$$
6	4 5	dom2d	$$⊢A=A\to \left(B\in V\to A\preccurlyeq B\right)$$
7	3 6	ax-mp	$$⊢B\in V\to A\preccurlyeq B$$