Theorem frege124

Description: If X is a result of an application of the single-valued procedure R to Y and if M follows Y in the R -sequence, then M belongs to the R -sequence beginning with X . Proposition 124 of Frege1879 p. 80. (Contributed by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege123.x	|- X e. U
	frege123.y	|- Y e. V
	frege124.m	|- M e. W
	frege124.r	|- R e. S
Assertion	frege124	|- ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege123.x	|- X e. U
2		frege123.y	|- Y e. V
3		frege124.m	|- M e. W
4		frege124.r	|- R e. S
5	1 2 3 4	frege110	|- ( A. a ( Y R a -> X ( ( t+ ` R ) u. _I ) a ) -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) )
6	1 2	frege123	|- ( ( A. a ( Y R a -> X ( ( t+ ` R ) u. _I ) a ) -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) ) -> ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) ) ) )
7	5 6	ax-mp	|- ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) ) )