Metamath Proof Explorer


Theorem frege126

Description: If M follows Y in the R -sequence and if the procedure R is single-valued, then every result of an application of the procedure R to Y belongs to the R -sequence beginning with M or precedes M in the R -sequence. Proposition 126 of Frege1879 p. 81. (Contributed by RP, 9-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 frege126
|- ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) )

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 3 1 frege114
 |-  ( X ( ( t+ ` R ) u. _I ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) )
6 1 2 3 4 frege125
 |-  ( ( X ( ( t+ ` R ) u. _I ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) -> ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) ) )
7 5 6 ax-mp
 |-  ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) )