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Metamath Proof Explorer

Theorem erref

Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of Enderton p. 56. (Contributed by Mario Carneiro, 6-May-2013) (Revised by Mario Carneiro, 12-Aug-2015)

Ref Expression
Hypotheses ersymb.1 
|- ( ph -> R Er X )
erref.2 
|- ( ph -> A e. X )
Assertion erref 
|- ( ph -> A R A )

Proof

Step Hyp Ref Expression
1 ersymb.1 
 |-  ( ph -> R Er X )
2 erref.2 
 |-  ( ph -> A e. X )
3 erdm 
 |-  ( R Er X -> dom R = X )
4 1 3 syl 
 |-  ( ph -> dom R = X )
5 2 4 eleqtrrd 
 |-  ( ph -> A e. dom R )
6 eldmg 
 |-  ( A e. X -> ( A e. dom R <-> E. x A R x ) )
7 2 6 syl 
 |-  ( ph -> ( A e. dom R <-> E. x A R x ) )
8 5 7 mpbid 
 |-  ( ph -> E. x A R x )
9 1 adantr 
 |-  ( ( ph /\ A R x ) -> R Er X )
10 simpr 
 |-  ( ( ph /\ A R x ) -> A R x )
11 9 10 10 ertr4d 
 |-  ( ( ph /\ A R x ) -> A R A )
12 8 11 exlimddv 
 |-  ( ph -> A R A )