Theorem dibeldmN

Description: Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		dibfn.b	$$⊢B={\mathrm{Base}}_K$$
2		dibfn.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dibfn.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		dibfn.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
5		eqid	$$⊢\mathrm{DIsoA}\left(K\right)\left(W\right)=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
6	3 5 4	dibdiadm	$$⊢\left(K\in V\wedge W\in H\right)\to \mathrm{dom}I=\mathrm{dom}\mathrm{DIsoA}\left(K\right)\left(W\right)$$
7	6	eleq2d	$$⊢\left(K\in V\wedge W\in H\right)\to \left(X\in \mathrm{dom}I\leftrightarrow X\in \mathrm{dom}\mathrm{DIsoA}\left(K\right)\left(W\right)\right)$$
8	1 2 3 5	diaeldm	$$⊢\left(K\in V\wedge W\in H\right)\to \left(X\in \mathrm{dom}\mathrm{DIsoA}\left(K\right)\left(W\right)\leftrightarrow \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)$$
9	7 8	bitrd	$$⊢\left(K\in V\wedge W\in H\right)\to \left(X\in \mathrm{dom}I\leftrightarrow \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)$$