Theorem mptsuppd

Description: The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019) (Revised by AV, 28-May-2019)

	Ref	Expression
Hypotheses	mptsuppdifd.f	$$⊢F=\left(x\in A⟼B\right)$$
	mptsuppdifd.a	$$⊢\phi \to A\in V$$
	mptsuppdifd.z	$$⊢\phi \to Z\in W$$
	mptsuppd.b	$$⊢\left(\phi \wedge x\in A\right)\to B\in U$$
Assertion	mptsuppd	$$⊢\phi \to F\mathrm{supp}Z=\left\{x\in A|B\ne Z\right\}$$

Proof

Step	Hyp	Ref	Expression
1		mptsuppdifd.f	$$⊢F=\left(x\in A⟼B\right)$$
2		mptsuppdifd.a	$$⊢\phi \to A\in V$$
3		mptsuppdifd.z	$$⊢\phi \to Z\in W$$
4		mptsuppd.b	$$⊢\left(\phi \wedge x\in A\right)\to B\in U$$
5	1 2 3	mptsuppdifd	$$⊢\phi \to F\mathrm{supp}Z=\left\{x\in A|B\in \left(\mathrm{V}\setminus \left\{Z\right\}\right)\right\}$$
6	4	elexd	$$⊢\left(\phi \wedge x\in A\right)\to B\in \mathrm{V}$$
7	6	biantrurd	$$⊢\left(\phi \wedge x\in A\right)\to \left(B\ne Z\leftrightarrow \left(B\in \mathrm{V}\wedge B\ne Z\right)\right)$$
8		eldifsn	$$⊢B\in \left(\mathrm{V}\setminus \left\{Z\right\}\right)\leftrightarrow \left(B\in \mathrm{V}\wedge B\ne Z\right)$$
9	7 8	syl6rbbr	$$⊢\left(\phi \wedge x\in A\right)\to \left(B\in \left(\mathrm{V}\setminus \left\{Z\right\}\right)\leftrightarrow B\ne Z\right)$$
10	9	rabbidva	$$⊢\phi \to \left\{x\in A|B\in \left(\mathrm{V}\setminus \left\{Z\right\}\right)\right\}=\left\{x\in A|B\ne Z\right\}$$
11	5 10	eqtrd	$$⊢\phi \to F\mathrm{supp}Z=\left\{x\in A|B\ne Z\right\}$$