Theorem psrbagfsuppOLD

Description: Obsolete version of psrbagfsupp as of 7-Aug-2024. (Contributed by Stefan O'Rear, 9-Mar-2015) (Revised by AV, 18-Jul-2019) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	psrbag.d	|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
Assertion	psrbagfsuppOLD	|- ( ( X e. D /\ I e. V ) -> X finSupp 0 )

Proof

Step	Hyp	Ref	Expression
1		psrbag.d	|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
2	1	psrbag	|- ( I e. V -> ( X e. D <-> ( X : I --> NN0 /\ ( `' X " NN ) e. Fin ) ) )
3	2	biimpac	|- ( ( X e. D /\ I e. V ) -> ( X : I --> NN0 /\ ( `' X " NN ) e. Fin ) )
4	3	simprd	|- ( ( X e. D /\ I e. V ) -> ( `' X " NN ) e. Fin )
5		simpr	|- ( ( X e. D /\ I e. V ) -> I e. V )
6	1	psrbagfOLD	|- ( ( I e. V /\ X e. D ) -> X : I --> NN0 )
7	6	ancoms	|- ( ( X e. D /\ I e. V ) -> X : I --> NN0 )
8		frnnn0fsupp	|- ( ( I e. V /\ X : I --> NN0 ) -> ( X finSupp 0 <-> ( `' X " NN ) e. Fin ) )
9	5 7 8	syl2anc	|- ( ( X e. D /\ I e. V ) -> ( X finSupp 0 <-> ( `' X " NN ) e. Fin ) )
10	4 9	mpbird	|- ( ( X e. D /\ I e. V ) -> X finSupp 0 )