Theorem frege118 38275

Description: Simplified application of one direction of dffrege115 38272. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege116.x	|- X e. U
frege118.y	|- Y e. V

Assertion

Ref	Expression
frege118	|- ( Fun `' `' R -> ( Y R X -> A. a ( Y R a -> a = X ) ) )

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

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

Proof of Theorem frege118

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege118.y	. . . 4 |- Y e. V
2	1	frege58c 38215	. . 3 |- ( A. b ( b R X -> A. a ( b R a -> a = X ) ) -> [. Y / b ]. ( b R X -> A. a ( b R a -> a = X ) ) )
3		sbcimg 3477	. . . . 5 |- ( Y e. V -> ( [. Y / b ]. ( b R X -> A. a ( b R a -> a = X ) ) <-> ( [. Y / b ]. b R X -> [. Y / b ]. A. a ( b R a -> a = X ) ) ) )
4	1, 3	ax-mp 5	. . . 4 |- ( [. Y / b ]. ( b R X -> A. a ( b R a -> a = X ) ) <-> ( [. Y / b ]. b R X -> [. Y / b ]. A. a ( b R a -> a = X ) ) )
5		sbcbr1g 4709	. . . . . . 7 |- ( Y e. V -> ( [. Y / b ]. b R X <-> [_ Y / b ]_ b R X ) )
6	1, 5	ax-mp 5	. . . . . 6 |- ( [. Y / b ]. b R X <-> [_ Y / b ]_ b R X )
7		csbvarg 4003	. . . . . . . 8 |- ( Y e. V -> [_ Y / b ]_ b = Y )
8	1, 7	ax-mp 5	. . . . . . 7 |- [_ Y / b ]_ b = Y
9	8	breq1i 4660	. . . . . 6 |- ( [_ Y / b ]_ b R X <-> Y R X )
10	6, 9	bitri 264	. . . . 5 |- ( [. Y / b ]. b R X <-> Y R X )
11		sbcal 3485	. . . . . 6 |- ( [. Y / b ]. A. a ( b R a -> a = X ) <-> A. a [. Y / b ]. ( b R a -> a = X ) )
12		sbcimg 3477	. . . . . . . . 9 |- ( Y e. V -> ( [. Y / b ]. ( b R a -> a = X ) <-> ( [. Y / b ]. b R a -> [. Y / b ]. a = X ) ) )
13	1, 12	ax-mp 5	. . . . . . . 8 |- ( [. Y / b ]. ( b R a -> a = X ) <-> ( [. Y / b ]. b R a -> [. Y / b ]. a = X ) )
14		sbcbr1g 4709	. . . . . . . . . . 11 |- ( Y e. V -> ( [. Y / b ]. b R a <-> [_ Y / b ]_ b R a ) )
15	1, 14	ax-mp 5	. . . . . . . . . 10 |- ( [. Y / b ]. b R a <-> [_ Y / b ]_ b R a )
16	8	breq1i 4660	. . . . . . . . . 10 |- ( [_ Y / b ]_ b R a <-> Y R a )
17	15, 16	bitri 264	. . . . . . . . 9 |- ( [. Y / b ]. b R a <-> Y R a )
18		sbcg 3503	. . . . . . . . . 10 |- ( Y e. V -> ( [. Y / b ]. a = X <-> a = X ) )
19	1, 18	ax-mp 5	. . . . . . . . 9 |- ( [. Y / b ]. a = X <-> a = X )
20	17, 19	imbi12i 340	. . . . . . . 8 |- ( ( [. Y / b ]. b R a -> [. Y / b ]. a = X ) <-> ( Y R a -> a = X ) )
21	13, 20	bitri 264	. . . . . . 7 |- ( [. Y / b ]. ( b R a -> a = X ) <-> ( Y R a -> a = X ) )
22	21	albii 1747	. . . . . 6 |- ( A. a [. Y / b ]. ( b R a -> a = X ) <-> A. a ( Y R a -> a = X ) )
23	11, 22	bitri 264	. . . . 5 |- ( [. Y / b ]. A. a ( b R a -> a = X ) <-> A. a ( Y R a -> a = X ) )
24	10, 23	imbi12i 340	. . . 4 |- ( ( [. Y / b ]. b R X -> [. Y / b ]. A. a ( b R a -> a = X ) ) <-> ( Y R X -> A. a ( Y R a -> a = X ) ) )
25	4, 24	bitri 264	. . 3 |- ( [. Y / b ]. ( b R X -> A. a ( b R a -> a = X ) ) <-> ( Y R X -> A. a ( Y R a -> a = X ) ) )
26	2, 25	sylib 208	. 2 |- ( A. b ( b R X -> A. a ( b R a -> a = X ) ) -> ( Y R X -> A. a ( Y R a -> a = X ) ) )
27		frege116.x	. . 3 |- X e. U
28	27	frege117 38274	. 2 |- ( ( A. b ( b R X -> A. a ( b R a -> a = X ) ) -> ( Y R X -> A. a ( Y R a -> a = X ) ) ) -> ( Fun `' `' R -> ( Y R X -> A. a ( Y R a -> a = X ) ) ) )
29	26, 28	ax-mp 5	1 |- ( Fun `' `' R -> ( Y R X -> A. a ( Y R a -> a = X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   [.wsbc 3435   [_csb 3533   class class class wbr 4653   `'ccnv 5113   Fun wfun 5882

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-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:  frege119  38276