Theorem frege123 38280

Description: Lemma for frege124 38281. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege123.x	|- X e. U
frege123.y	|- Y e. V

Assertion

Ref	Expression
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 ) ) ) )

Distinct variable groups:   R, a   X, a   Y, a

Allowed substitution hints:   U( a)   M( a)   V( a)

Proof of Theorem frege123

Step	Hyp	Ref	Expression
1		frege123.x	. . . 4 |- X e. U
2		frege123.y	. . . 4 |- Y e. V
3		vex 3203	. . . 4 |- a e. _V
4	1, 2, 3	frege122 38279	. . 3 |- ( Fun `' `' R -> ( Y R X -> ( Y R a -> X ( ( t+ ` R ) u. _I ) a ) ) )
5	4	alrimdv 1857	. 2 |- ( Fun `' `' R -> ( Y R X -> A. a ( Y R a -> X ( ( t+ ` R ) u. _I ) a ) ) )
6		frege19 38118	. 2 |- ( ( Fun `' `' R -> ( Y R X -> A. a ( Y R a -> X ( ( t+ ` R ) u. _I ) a ) ) ) -> ( ( 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 5	1 |- ( ( 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 ) ) ) )

Syntax hints:   -> wi 4   A.wal 1481   e. wcel 1990   _Vcvv 3200   u. cun 3572   class class class wbr 4653   _I cid 5023   `'ccnv 5113   Fun wfun 5882   ` cfv 5888   t+ctcl 13724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege52a 38151  ax-frege58b 38195

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  frege124  38281