Theorem opabiotadm

Description: Define a function whose value is "the unique y such that ph ( x , y ) ". (Contributed by NM, 16-Nov-2013)

	Ref	Expression
Hypothesis	opabiota.1	$$⊢F=\left\{⟨x,y⟩|\left\{y|\phi \right\}=\left\{y\right\}\right\}$$
Assertion	opabiotadm	$$⊢\mathrm{dom}F=\left\{x|\exists !y\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		opabiota.1	$$⊢F=\left\{⟨x,y⟩|\left\{y|\phi \right\}=\left\{y\right\}\right\}$$
2		dmopab	$$⊢\mathrm{dom}\left\{⟨x,y⟩|\left\{y|\phi \right\}=\left\{y\right\}\right\}=\left\{x|\exists y\left\{y|\phi \right\}=\left\{y\right\}\right\}$$
3	1	dmeqi	$$⊢\mathrm{dom}F=\mathrm{dom}\left\{⟨x,y⟩|\left\{y|\phi \right\}=\left\{y\right\}\right\}$$
4		euabsn	$$⊢\exists !y\phi \leftrightarrow \exists y\left\{y|\phi \right\}=\left\{y\right\}$$
5	4	abbii	$$⊢\left\{x|\exists !y\phi \right\}=\left\{x|\exists y\left\{y|\phi \right\}=\left\{y\right\}\right\}$$
6	2 3 5	3eqtr4i	$$⊢\mathrm{dom}F=\left\{x|\exists !y\phi \right\}$$