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Metamath Proof Explorer

Theorem caofid0l

Description: Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014)

Ref Expression
Hypotheses caofref.1 
|- ( ph -> A e. V )
caofref.2 
|- ( ph -> F : A --> S )
caofid0.3 
|- ( ph -> B e. W )
caofid0l.5 
|- ( ( ph /\ x e. S ) -> ( B R x ) = x )
Assertion caofid0l 
|- ( ph -> ( ( A X. { B } ) oF R F ) = F )

Proof

Step Hyp Ref Expression
1 caofref.1 
 |-  ( ph -> A e. V )
2 caofref.2 
 |-  ( ph -> F : A --> S )
3 caofid0.3 
 |-  ( ph -> B e. W )
4 caofid0l.5 
 |-  ( ( ph /\ x e. S ) -> ( B R x ) = x )
5 fnconstg 
 |-  ( B e. W -> ( A X. { B } ) Fn A )
6 3 5 syl 
 |-  ( ph -> ( A X. { B } ) Fn A )
7 2 ffnd 
 |-  ( ph -> F Fn A )
8 fvconst2g 
 |-  ( ( B e. W /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
9 3 8 sylan 
 |-  ( ( ph /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
10 eqidd 
 |-  ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
11 4 ralrimiva 
 |-  ( ph -> A. x e. S ( B R x ) = x )
12 2 ffvelcdmda 
 |-  ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
13 oveq2 
 |-  ( x = ( F ` w ) -> ( B R x ) = ( B R ( F ` w ) ) )
14 id 
 |-  ( x = ( F ` w ) -> x = ( F ` w ) )
15 13 14 eqeq12d 
 |-  ( x = ( F ` w ) -> ( ( B R x ) = x <-> ( B R ( F ` w ) ) = ( F ` w ) ) )
16 15 rspccva 
 |-  ( ( A. x e. S ( B R x ) = x /\ ( F ` w ) e. S ) -> ( B R ( F ` w ) ) = ( F ` w ) )
17 11 12 16 syl2an2r 
 |-  ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = ( F ` w ) )
18 1 6 7 7 9 10 17 offveq 
 |-  ( ph -> ( ( A X. { B } ) oF R F ) = F )