Description: Lemma for frege133 . Proposition 132 of Frege1879 p. 86. (Contributed by RP, 9-Jul-2020) (Proof modification is discouraged.)
Ref | Expression | ||
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Hypotheses | frege130.m | |- M e. U |
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frege130.r | |- R e. V |
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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 ) ) ) ) ) |
Step | Hyp | Ref | Expression |
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1 | frege130.m | |- M e. U |
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2 | frege130.r | |- R e. V |
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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 ) ) ) ) ) ) |
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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 ) ) ) ) ) |