Theorem eqfunressuc

Description: Law for equality of restriction to successors. This is primarily useful when X is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021)

	Ref	Expression
Assertion	eqfunressuc	|- ( ( ( Fun F /\ Fun G ) /\ ( F |` X ) = ( G |` X ) /\ ( X e. dom F /\ X e. dom G /\ ( F ` X ) = ( G ` X ) ) ) -> ( F |` suc X ) = ( G |` suc X ) )

Proof

Step	Hyp	Ref	Expression
1		eqfunresadj	|- ( ( ( Fun F /\ Fun G ) /\ ( F |` X ) = ( G |` X ) /\ ( X e. dom F /\ X e. dom G /\ ( F ` X ) = ( G ` X ) ) ) -> ( F |` ( X u. { X } ) ) = ( G |` ( X u. { X } ) ) )
2		df-suc	|- suc X = ( X u. { X } )
3	2	reseq2i	|- ( F |` suc X ) = ( F |` ( X u. { X } ) )
4	2	reseq2i	|- ( G |` suc X ) = ( G |` ( X u. { X } ) )
5	1 3 4	3eqtr4g	|- ( ( ( Fun F /\ Fun G ) /\ ( F |` X ) = ( G |` X ) /\ ( X e. dom F /\ X e. dom G /\ ( F ` X ) = ( G ` X ) ) ) -> ( F |` suc X ) = ( G |` suc X ) )