Theorem ind1a

Description: Value of the indicator function where it is 1 . (Contributed by Thierry Arnoux, 22-Aug-2017)

	Ref	Expression
Assertion	ind1a	$$⊢\left(O\in V\wedge A\subseteq O\wedge X\in O\right)\to \left({𝟙}_O\left(A\right)\left(X\right)=1\leftrightarrow X\in A\right)$$

Proof

Step	Hyp	Ref	Expression
1		indfval	$$⊢\left(O\in V\wedge A\subseteq O\wedge X\in O\right)\to {𝟙}_O\left(A\right)\left(X\right)=if\left(X\in A,1,0\right)$$
2	1	eqeq1d	$$⊢\left(O\in V\wedge A\subseteq O\wedge X\in O\right)\to \left({𝟙}_O\left(A\right)\left(X\right)=1\leftrightarrow if\left(X\in A,1,0\right)=1\right)$$
3		eqid	$$⊢1=1$$
4	3	biantru	$$⊢X\in A\leftrightarrow \left(X\in A\wedge 1=1\right)$$
5		ax-1ne0	$$⊢1\ne 0$$
6	5	neii	$$⊢\neg 1=0$$
7	6	biorfi	$$⊢\left(X\in A\wedge 1=1\right)\leftrightarrow \left(\left(X\in A\wedge 1=1\right)\vee 1=0\right)$$
8	6	bianfi	$$⊢1=0\leftrightarrow \left(\neg X\in A\wedge 1=0\right)$$
9	8	orbi2i	$$⊢\left(\left(X\in A\wedge 1=1\right)\vee 1=0\right)\leftrightarrow \left(\left(X\in A\wedge 1=1\right)\vee \left(\neg X\in A\wedge 1=0\right)\right)$$
10	4 7 9	3bitri	$$⊢X\in A\leftrightarrow \left(\left(X\in A\wedge 1=1\right)\vee \left(\neg X\in A\wedge 1=0\right)\right)$$
11		eqif	$$⊢1=if\left(X\in A,1,0\right)\leftrightarrow \left(\left(X\in A\wedge 1=1\right)\vee \left(\neg X\in A\wedge 1=0\right)\right)$$
12		eqcom	$$⊢1=if\left(X\in A,1,0\right)\leftrightarrow if\left(X\in A,1,0\right)=1$$
13	10 11 12	3bitr2ri	$$⊢if\left(X\in A,1,0\right)=1\leftrightarrow X\in A$$
14	2 13	bitrdi	$$⊢\left(O\in V\wedge A\subseteq O\wedge X\in O\right)\to \left({𝟙}_O\left(A\right)\left(X\right)=1\leftrightarrow X\in A\right)$$