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Metamath Proof Explorer

Theorem caofid1

Description: Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014)

Ref Expression
Hypotheses caofref.1 
|- ( ph -> A e. V )
caofref.2 
|- ( ph -> F : A --> S )
caofid0.3 
|- ( ph -> B e. W )
caofid1.4 
|- ( ph -> C e. X )
caofid1.5 
|- ( ( ph /\ x e. S ) -> ( x R B ) = C )
Assertion caofid1 
|- ( ph -> ( F oF R ( A X. { B } ) ) = ( A X. { C } ) )

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 caofid1.4 
 |-  ( ph -> C e. X )
5 caofid1.5 
 |-  ( ( ph /\ x e. S ) -> ( x R B ) = C )
6 2 ffnd 
 |-  ( ph -> F Fn A )
7 fnconstg 
 |-  ( B e. W -> ( A X. { B } ) Fn A )
8 3 7 syl 
 |-  ( ph -> ( A X. { B } ) Fn A )
9 fnconstg 
 |-  ( C e. X -> ( A X. { C } ) Fn A )
10 4 9 syl 
 |-  ( ph -> ( A X. { C } ) Fn A )
11 eqidd 
 |-  ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
12 fvconst2g 
 |-  ( ( B e. W /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
13 3 12 sylan 
 |-  ( ( ph /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
14 5 ralrimiva 
 |-  ( ph -> A. x e. S ( x R B ) = C )
15 2 ffvelcdmda 
 |-  ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
16 oveq1 
 |-  ( x = ( F ` w ) -> ( x R B ) = ( ( F ` w ) R B ) )
17 16 eqeq1d 
 |-  ( x = ( F ` w ) -> ( ( x R B ) = C <-> ( ( F ` w ) R B ) = C ) )
18 17 rspccva 
 |-  ( ( A. x e. S ( x R B ) = C /\ ( F ` w ) e. S ) -> ( ( F ` w ) R B ) = C )
19 14 15 18 syl2an2r 
 |-  ( ( ph /\ w e. A ) -> ( ( F ` w ) R B ) = C )
20 fvconst2g 
 |-  ( ( C e. X /\ w e. A ) -> ( ( A X. { C } ) ` w ) = C )
21 4 20 sylan 
 |-  ( ( ph /\ w e. A ) -> ( ( A X. { C } ) ` w ) = C )
22 19 21 eqtr4d 
 |-  ( ( ph /\ w e. A ) -> ( ( F ` w ) R B ) = ( ( A X. { C } ) ` w ) )
23 1 6 8 10 11 13 22 offveq 
 |-  ( ph -> ( F oF R ( A X. { B } ) ) = ( A X. { C } ) )