Metamath Proof Explorer


Theorem frege122

Description: If X is a result of an application of the single-valued procedure R to Y , then every result of an application of the procedure R to Y belongs to the R -sequence beginning with X . Proposition 122 of Frege1879 p. 79. (Contributed by RP, 8-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypotheses frege116.x
|- X e. U
frege118.y
|- Y e. V
frege120.a
|- A e. W
Assertion frege122
|- ( Fun `' `' R -> ( Y R X -> ( Y R A -> X ( ( t+ ` R ) u. _I ) A ) ) )

Proof

Step Hyp Ref Expression
1 frege116.x
 |-  X e. U
2 frege118.y
 |-  Y e. V
3 frege120.a
 |-  A e. W
4 3 frege112
 |-  ( A = X -> X ( ( t+ ` R ) u. _I ) A )
5 1 2 3 frege121
 |-  ( ( A = X -> X ( ( t+ ` R ) u. _I ) A ) -> ( Fun `' `' R -> ( Y R X -> ( Y R A -> X ( ( t+ ` R ) u. _I ) A ) ) ) )
6 4 5 ax-mp
 |-  ( Fun `' `' R -> ( Y R X -> ( Y R A -> X ( ( t+ ` R ) u. _I ) A ) ) )