Metamath Proof Explorer


Theorem frege132

Description: Lemma for frege133 . Proposition 132 of Frege1879 p. 86. (Contributed by RP, 9-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypotheses frege130.m
|- M e. U
frege130.r
|- R e. V
Assertion frege132
|- ( ( R hereditary ( ( `' ( t+ ` R ) " { M } ) u. ( ( ( t+ ` R ) u. _I ) " { M } ) ) -> ( X ( t+ ` R ) M -> ( X ( t+ ` R ) Y -> ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) ) ) ) -> ( Fun `' `' R -> ( X ( t+ ` R ) M -> ( X ( t+ ` R ) Y -> ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) ) ) ) )

Proof

Step Hyp Ref Expression
1 frege130.m
 |-  M e. U
2 frege130.r
 |-  R e. V
3 1 2 frege131
 |-  ( Fun `' `' R -> R hereditary ( ( `' ( t+ ` R ) " { M } ) u. ( ( ( t+ ` R ) u. _I ) " { M } ) ) )
4 frege9
 |-  ( ( Fun `' `' R -> R hereditary ( ( `' ( t+ ` R ) " { M } ) u. ( ( ( t+ ` R ) u. _I ) " { M } ) ) ) -> ( ( R hereditary ( ( `' ( t+ ` R ) " { M } ) u. ( ( ( t+ ` R ) u. _I ) " { M } ) ) -> ( X ( t+ ` R ) M -> ( X ( t+ ` R ) Y -> ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) ) ) ) -> ( Fun `' `' R -> ( X ( t+ ` R ) M -> ( X ( t+ ` R ) Y -> ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) ) ) ) ) )
5 3 4 ax-mp
 |-  ( ( R hereditary ( ( `' ( t+ ` R ) " { M } ) u. ( ( ( t+ ` R ) u. _I ) " { M } ) ) -> ( X ( t+ ` R ) M -> ( X ( t+ ` R ) Y -> ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) ) ) ) -> ( Fun `' `' R -> ( X ( t+ ` R ) M -> ( X ( t+ ` R ) Y -> ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) ) ) ) )