Theorem diafn

Description: Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-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	diafn	$$⊢\left(K\in V\wedge W\in H\right)\to IFn\left\{x\in B|x\underset{\dot{}}{\le }W\right\}$$

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		fvex	$$⊢\mathrm{LTrn}\left(K\right)\left(W\right)\in \mathrm{V}$$
6	5	rabex	$$⊢\left\{f\in \mathrm{LTrn}\left(K\right)\left(W\right)|\mathrm{trL}\left(K\right)\left(W\right)\left(f\right)\underset{\dot{}}{\le }y\right\}\in \mathrm{V}$$
7		eqid	$$⊢\left(y\in \left\{x\in B|x\underset{\dot{}}{\le }W\right\}⟼\left\{f\in \mathrm{LTrn}\left(K\right)\left(W\right)|\mathrm{trL}\left(K\right)\left(W\right)\left(f\right)\underset{\dot{}}{\le }y\right\}\right)=\left(y\in \left\{x\in B|x\underset{\dot{}}{\le }W\right\}⟼\left\{f\in \mathrm{LTrn}\left(K\right)\left(W\right)|\mathrm{trL}\left(K\right)\left(W\right)\left(f\right)\underset{\dot{}}{\le }y\right\}\right)$$
8	6 7	fnmpti	$$⊢\left(y\in \left\{x\in B|x\underset{\dot{}}{\le }W\right\}⟼\left\{f\in \mathrm{LTrn}\left(K\right)\left(W\right)|\mathrm{trL}\left(K\right)\left(W\right)\left(f\right)\underset{\dot{}}{\le }y\right\}\right)Fn\left\{x\in B|x\underset{\dot{}}{\le }W\right\}$$
9		eqid	$$⊢\mathrm{LTrn}\left(K\right)\left(W\right)=\mathrm{LTrn}\left(K\right)\left(W\right)$$
10		eqid	$$⊢\mathrm{trL}\left(K\right)\left(W\right)=\mathrm{trL}\left(K\right)\left(W\right)$$
11	1 2 3 9 10 4	diafval	$$⊢\left(K\in V\wedge W\in H\right)\to I=\left(y\in \left\{x\in B|x\underset{\dot{}}{\le }W\right\}⟼\left\{f\in \mathrm{LTrn}\left(K\right)\left(W\right)|\mathrm{trL}\left(K\right)\left(W\right)\left(f\right)\underset{\dot{}}{\le }y\right\}\right)$$
12	11	fneq1d	$$⊢\left(K\in V\wedge W\in H\right)\to \left(IFn\left\{x\in B|x\underset{\dot{}}{\le }W\right\}\leftrightarrow \left(y\in \left\{x\in B|x\underset{\dot{}}{\le }W\right\}⟼\left\{f\in \mathrm{LTrn}\left(K\right)\left(W\right)|\mathrm{trL}\left(K\right)\left(W\right)\left(f\right)\underset{\dot{}}{\le }y\right\}\right)Fn\left\{x\in B|x\underset{\dot{}}{\le }W\right\}\right)$$
13	8 12	mpbiri	$$⊢\left(K\in V\wedge W\in H\right)\to IFn\left\{x\in B|x\underset{\dot{}}{\le }W\right\}$$