Theorem diaeldm

Description: Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013)

	Ref	Expression
Hypotheses	diafn.b	$$⊢B={\mathrm{Base}}_K$$
	diafn.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	diafn.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	diafn.i	$$⊢I=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
Assertion	diaeldm	$$⊢\left(K\in V\wedge W\in H\right)\to \left(X\in \mathrm{dom}I\leftrightarrow \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		diafn.b	$$⊢B={\mathrm{Base}}_K$$
2		diafn.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		diafn.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		diafn.i	$$⊢I=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
5	1 2 3 4	diadm	$$⊢\left(K\in V\wedge W\in H\right)\to \mathrm{dom}I=\left\{x\in B|x\underset{\dot{}}{\le }W\right\}$$
6	5	eleq2d	$$⊢\left(K\in V\wedge W\in H\right)\to \left(X\in \mathrm{dom}I\leftrightarrow X\in \left\{x\in B|x\underset{\dot{}}{\le }W\right\}\right)$$
7		breq1	$$⊢x=X\to \left(x\underset{\dot{}}{\le }W\leftrightarrow X\underset{\dot{}}{\le }W\right)$$
8	7	elrab	$$⊢X\in \left\{x\in B|x\underset{\dot{}}{\le }W\right\}\leftrightarrow \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)$$
9	6 8	bitrdi	$$⊢\left(K\in V\wedge W\in H\right)\to \left(X\in \mathrm{dom}I\leftrightarrow \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)$$