Theorem dibdmN

Description: Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-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	dibdmN	$$⊢\left(K\in V\wedge W\in H\right)\to \mathrm{dom}I=\left\{x\in B|x\underset{\dot{}}{\le }W\right\}$$

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	1 2 3 4	dibfnN	$$⊢\left(K\in V\wedge W\in H\right)\to IFn\left\{x\in B|x\underset{\dot{}}{\le }W\right\}$$
6		fndm	$$⊢IFn\left\{x\in B|x\underset{\dot{}}{\le }W\right\}\to \mathrm{dom}I=\left\{x\in B|x\underset{\dot{}}{\le }W\right\}$$
7	5 6	syl	$$⊢\left(K\in V\wedge W\in H\right)\to \mathrm{dom}I=\left\{x\in B|x\underset{\dot{}}{\le }W\right\}$$