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Metamath Proof Explorer

Theorem fvco2

Description: Value of a function composition. Similar to second part of Theorem 3H of Enderton p. 47. (Contributed by NM, 9-Oct-2004) (Proof shortened by Andrew Salmon, 22-Oct-2011) (Revised by Stefan O'Rear, 16-Oct-2014)

Ref Expression
Assertion fvco2 
|- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )

Proof

Step Hyp Ref Expression
1 imaco 
 |-  ( ( F o. G ) " { X } ) = ( F " ( G " { X } ) )
2 fnsnfv 
 |-  ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
3 2 imaeq2d 
 |-  ( ( G Fn A /\ X e. A ) -> ( F " { ( G ` X ) } ) = ( F " ( G " { X } ) ) )
4 1 3 eqtr4id 
 |-  ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) " { X } ) = ( F " { ( G ` X ) } ) )
5 4 eleq2d 
 |-  ( ( G Fn A /\ X e. A ) -> ( x e. ( ( F o. G ) " { X } ) <-> x e. ( F " { ( G ` X ) } ) ) )
6 5 iotabidv 
 |-  ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7 dffv3 
 |-  ( ( F o. G ) ` X ) = ( iota x x e. ( ( F o. G ) " { X } ) )
8 dffv3 
 |-  ( F ` ( G ` X ) ) = ( iota x x e. ( F " { ( G ` X ) } ) )
9 6 7 8 3eqtr4g 
 |-  ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )