Theorem seppsepf

Description: If two sets are precisely separated by a continuous function, then they are separated by the continuous function. (Contributed by Zhi Wang, 9-Sep-2024)

	Ref	Expression
Hypothesis	seppsepf.1	$$⊢\phi \to \exists f\in \left(J\mathrm{Cn}\mathrm{II}\right)\left(S=f^{-1}\left[\left\{0\right\}\right]\wedge T=f^{-1}\left[\left\{1\right\}\right]\right)$$
Assertion	seppsepf	$$⊢\phi \to \exists f\in \left(J\mathrm{Cn}\mathrm{II}\right)\left(S\subseteq f^{-1}\left[\left\{0\right\}\right]\wedge T\subseteq f^{-1}\left[\left\{1\right\}\right]\right)$$

Proof

Step	Hyp	Ref	Expression
1		seppsepf.1	$$⊢\phi \to \exists f\in \left(J\mathrm{Cn}\mathrm{II}\right)\left(S=f^{-1}\left[\left\{0\right\}\right]\wedge T=f^{-1}\left[\left\{1\right\}\right]\right)$$
2		eqimss	$$⊢S=f^{-1}\left[\left\{0\right\}\right]\to S\subseteq f^{-1}\left[\left\{0\right\}\right]$$
3		eqimss	$$⊢T=f^{-1}\left[\left\{1\right\}\right]\to T\subseteq f^{-1}\left[\left\{1\right\}\right]$$
4	2 3	anim12i	$$⊢\left(S=f^{-1}\left[\left\{0\right\}\right]\wedge T=f^{-1}\left[\left\{1\right\}\right]\right)\to \left(S\subseteq f^{-1}\left[\left\{0\right\}\right]\wedge T\subseteq f^{-1}\left[\left\{1\right\}\right]\right)$$
5	4	reximi	$$⊢\exists f\in \left(J\mathrm{Cn}\mathrm{II}\right)\left(S=f^{-1}\left[\left\{0\right\}\right]\wedge T=f^{-1}\left[\left\{1\right\}\right]\right)\to \exists f\in \left(J\mathrm{Cn}\mathrm{II}\right)\left(S\subseteq f^{-1}\left[\left\{0\right\}\right]\wedge T\subseteq f^{-1}\left[\left\{1\right\}\right]\right)$$
6	1 5	syl	$$⊢\phi \to \exists f\in \left(J\mathrm{Cn}\mathrm{II}\right)\left(S\subseteq f^{-1}\left[\left\{0\right\}\right]\wedge T\subseteq f^{-1}\left[\left\{1\right\}\right]\right)$$