Theorem funconstss

Description: Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007)

	Ref	Expression
Assertion	funconstss	$$⊢\left(\mathrm{Fun}F\wedge A\subseteq \mathrm{dom}F\right)\to \left(\forall x\in AF\left(x\right)=B\leftrightarrow A\subseteq F^{-1}\left[\left\{B\right\}\right]\right)$$

Proof

Step	Hyp	Ref	Expression
1		funimass4	$$⊢\left(\mathrm{Fun}F\wedge A\subseteq \mathrm{dom}F\right)\to \left(F\left[A\right]\subseteq \left\{B\right\}\leftrightarrow \forall x\in AF\left(x\right)\in \left\{B\right\}\right)$$
2		fvex	$$⊢F\left(x\right)\in \mathrm{V}$$
3	2	elsn	$$⊢F\left(x\right)\in \left\{B\right\}\leftrightarrow F\left(x\right)=B$$
4	3	ralbii	$$⊢\forall x\in AF\left(x\right)\in \left\{B\right\}\leftrightarrow \forall x\in AF\left(x\right)=B$$
5	1 4	bitr2di	$$⊢\left(\mathrm{Fun}F\wedge A\subseteq \mathrm{dom}F\right)\to \left(\forall x\in AF\left(x\right)=B\leftrightarrow F\left[A\right]\subseteq \left\{B\right\}\right)$$
6		funimass3	$$⊢\left(\mathrm{Fun}F\wedge A\subseteq \mathrm{dom}F\right)\to \left(F\left[A\right]\subseteq \left\{B\right\}\leftrightarrow A\subseteq F^{-1}\left[\left\{B\right\}\right]\right)$$
7	5 6	bitrd	$$⊢\left(\mathrm{Fun}F\wedge A\subseteq \mathrm{dom}F\right)\to \left(\forall x\in AF\left(x\right)=B\leftrightarrow A\subseteq F^{-1}\left[\left\{B\right\}\right]\right)$$