# Metamath Proof Explorer

## Theorem caofid2

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 )`
caofid2.5
`|- ( ( ph /\ x e. S ) -> ( B R x ) = C )`
Assertion caofid2
`|- ( ph -> ( ( A X. { B } ) oF R F ) = ( 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 caofid2.5
` |-  ( ( ph /\ x e. S ) -> ( B R x ) = C )`
6 fnconstg
` |-  ( B e. W -> ( A X. { B } ) Fn A )`
7 3 6 syl
` |-  ( ph -> ( A X. { B } ) Fn A )`
8 2 ffnd
` |-  ( ph -> F Fn A )`
9 fnconstg
` |-  ( C e. X -> ( A X. { C } ) Fn A )`
10 4 9 syl
` |-  ( ph -> ( A X. { C } ) Fn A )`
11 fvconst2g
` |-  ( ( B e. W /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )`
12 3 11 sylan
` |-  ( ( ph /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )`
13 eqidd
` |-  ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )`
14 5 ralrimiva
` |-  ( ph -> A. x e. S ( B R x ) = C )`
15 2 ffvelrnda
` |-  ( ( ph /\ w e. A ) -> ( F ` w ) e. S )`
16 oveq2
` |-  ( x = ( F ` w ) -> ( B R x ) = ( B R ( F ` w ) ) )`
17 16 eqeq1d
` |-  ( x = ( F ` w ) -> ( ( B R x ) = C <-> ( B R ( F ` w ) ) = C ) )`
18 17 rspccva
` |-  ( ( A. x e. S ( B R x ) = C /\ ( F ` w ) e. S ) -> ( B R ( F ` w ) ) = C )`
19 14 15 18 syl2an2r
` |-  ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = 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 ) -> ( B R ( F ` w ) ) = ( ( A X. { C } ) ` w ) )`
23 1 7 8 10 12 13 22 offveq
` |-  ( ph -> ( ( A X. { B } ) oF R F ) = ( A X. { C } ) )`