Theorem dibval2

Description: Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014)

	Ref	Expression
Hypotheses	dibval2.b	$$⊢B={\mathrm{Base}}_K$$
	dibval2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dibval2.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dibval2.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	dibval2.o	$$⊢\underset{\dot{}}{0}=\left(f\in T⟼{\mathrm{I}\upharpoonright }_B\right)$$
	dibval2.j	$$⊢J=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
	dibval2.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
Assertion	dibval2	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=J\left(X\right)\times \left\{\underset{\dot{}}{0}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dibval2.b	$$⊢B={\mathrm{Base}}_K$$
2		dibval2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dibval2.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		dibval2.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
5		dibval2.o	$$⊢\underset{\dot{}}{0}=\left(f\in T⟼{\mathrm{I}\upharpoonright }_B\right)$$
6		dibval2.j	$$⊢J=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
7		dibval2.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
8	1 2 3 6	diaeldm	$$⊢\left(K\in V\wedge W\in H\right)\to \left(X\in \mathrm{dom}J\leftrightarrow \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)$$
9	8	biimpar	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to X\in \mathrm{dom}J$$
10	1 3 4 5 6 7	dibval	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge X\in \mathrm{dom}J\right)\to I\left(X\right)=J\left(X\right)\times \left\{\underset{\dot{}}{0}\right\}$$
11	9 10	syldan	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=J\left(X\right)\times \left\{\underset{\dot{}}{0}\right\}$$