Theorem qsinxp

Description: Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015)

	Ref	Expression
Assertion	qsinxp	$$⊢R\left[A\right]\subseteq A\to A/R=A/\left(R\cap \left(A\times A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ecinxp	$$⊢\left(R\left[A\right]\subseteq A\wedge x\in A\right)\to \left[x\right]R=\left[x\right]\left(R\cap \left(A\times A\right)\right)$$
2	1	eqeq2d	$$⊢\left(R\left[A\right]\subseteq A\wedge x\in A\right)\to \left(y=\left[x\right]R\leftrightarrow y=\left[x\right]\left(R\cap \left(A\times A\right)\right)\right)$$
3	2	rexbidva	$$⊢R\left[A\right]\subseteq A\to \left(\exists x\in Ay=\left[x\right]R\leftrightarrow \exists x\in Ay=\left[x\right]\left(R\cap \left(A\times A\right)\right)\right)$$
4	3	abbidv	$$⊢R\left[A\right]\subseteq A\to \left\{y|\exists x\in Ay=\left[x\right]R\right\}=\left\{y|\exists x\in Ay=\left[x\right]\left(R\cap \left(A\times A\right)\right)\right\}$$
5		df-qs	$$⊢A/R=\left\{y|\exists x\in Ay=\left[x\right]R\right\}$$
6		df-qs	$$⊢A/\left(R\cap \left(A\times A\right)\right)=\left\{y|\exists x\in Ay=\left[x\right]\left(R\cap \left(A\times A\right)\right)\right\}$$
7	4 5 6	3eqtr4g	$$⊢R\left[A\right]\subseteq A\to A/R=A/\left(R\cap \left(A\times A\right)\right)$$