Metamath Proof Explorer

Theorem elno3

Description: Another condition for membership in No . (Contributed by Scott Fenton, 14-Apr-2012)

Ref Expression
Assertion elno3 
|- ( A e. No <-> ( A : dom A --> { 1o , 2o } /\ dom A e. On ) )

Proof

Step Hyp Ref Expression
1 3anan32 
 |-  ( ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) <-> ( ( Fun A /\ ran A C_ { 1o , 2o } ) /\ dom A e. On ) )
2 elno2 
 |-  ( A e. No <-> ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) )
3 df-f 
 |-  ( A : dom A --> { 1o , 2o } <-> ( A Fn dom A /\ ran A C_ { 1o , 2o } ) )
4 funfn 
 |-  ( Fun A <-> A Fn dom A )
5 4 anbi1i 
 |-  ( ( Fun A /\ ran A C_ { 1o , 2o } ) <-> ( A Fn dom A /\ ran A C_ { 1o , 2o } ) )
6 3 5 bitr4i 
 |-  ( A : dom A --> { 1o , 2o } <-> ( Fun A /\ ran A C_ { 1o , 2o } ) )
7 6 anbi1i 
 |-  ( ( A : dom A --> { 1o , 2o } /\ dom A e. On ) <-> ( ( Fun A /\ ran A C_ { 1o , 2o } ) /\ dom A e. On ) )
8 1 2 7 3bitr4i 
 |-  ( A e. No <-> ( A : dom A --> { 1o , 2o } /\ dom A e. On ) )