Theorem elno2

Description: An alternative condition for membership in No . (Contributed by Scott Fenton, 21-Mar-2012)

	Ref	Expression
Assertion	elno2	|- ( A e. No <-> ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) )

Proof

Step	Hyp	Ref	Expression
1		nofun	|- ( A e. No -> Fun A )
2		nodmon	|- ( A e. No -> dom A e. On )
3		norn	|- ( A e. No -> ran A C_ { 1o , 2o } )
4	1 2 3	3jca	|- ( A e. No -> ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) )
5		simp2	|- ( ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) -> dom A e. On )
6		simpl	|- ( ( Fun A /\ dom A e. On ) -> Fun A )
7	6	funfnd	|- ( ( Fun A /\ dom A e. On ) -> A Fn dom A )
8	7	anim1i	|- ( ( ( Fun A /\ dom A e. On ) /\ ran A C_ { 1o , 2o } ) -> ( A Fn dom A /\ ran A C_ { 1o , 2o } ) )
9	8	3impa	|- ( ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) -> ( A Fn dom A /\ ran A C_ { 1o , 2o } ) )
10		df-f	|- ( A : dom A --> { 1o , 2o } <-> ( A Fn dom A /\ ran A C_ { 1o , 2o } ) )
11	9 10	sylibr	|- ( ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) -> A : dom A --> { 1o , 2o } )
12		feq2	|- ( x = dom A -> ( A : x --> { 1o , 2o } <-> A : dom A --> { 1o , 2o } ) )
13	12	rspcev	|- ( ( dom A e. On /\ A : dom A --> { 1o , 2o } ) -> E. x e. On A : x --> { 1o , 2o } )
14	5 11 13	syl2anc	|- ( ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) -> E. x e. On A : x --> { 1o , 2o } )
15		elno	|- ( A e. No <-> E. x e. On A : x --> { 1o , 2o } )
16	14 15	sylibr	|- ( ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) -> A e. No )
17	4 16	impbii	|- ( A e. No <-> ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) )