Theorem frege120 38277

Description: Simplified application of one direction of dffrege115 38272. Proposition 120 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
frege120.a	|- A e. W

Assertion

Ref	Expression
frege120	|- ( Fun `' `' R -> ( Y R X -> ( Y R A -> A = X ) ) )

Proof of Theorem frege120

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege120.a	. . . 4 |- A e. W
2	1	frege58c 38215	. . 3 |- ( A. a ( Y R a -> a = X ) -> [. A / a ]. ( Y R a -> a = X ) )
3		sbcim1 3482	. . . 4 |- ( [. A / a ]. ( Y R a -> a = X ) -> ( [. A / a ]. Y R a -> [. A / a ]. a = X ) )
4		sbcbr2g 4710	. . . . . 6 |- ( A e. W -> ( [. A / a ]. Y R a <-> Y R [_ A / a ]_ a ) )
5	1, 4	ax-mp 5	. . . . 5 |- ( [. A / a ]. Y R a <-> Y R [_ A / a ]_ a )
6		csbvarg 4003	. . . . . . 7 |- ( A e. W -> [_ A / a ]_ a = A )
7	1, 6	ax-mp 5	. . . . . 6 |- [_ A / a ]_ a = A
8	7	breq2i 4661	. . . . 5 |- ( Y R [_ A / a ]_ a <-> Y R A )
9	5, 8	bitri 264	. . . 4 |- ( [. A / a ]. Y R a <-> Y R A )
10		sbceq1g 3988	. . . . . 6 |- ( A e. W -> ( [. A / a ]. a = X <-> [_ A / a ]_ a = X ) )
11	1, 10	ax-mp 5	. . . . 5 |- ( [. A / a ]. a = X <-> [_ A / a ]_ a = X )
12	7	eqeq1i 2627	. . . . 5 |- ( [_ A / a ]_ a = X <-> A = X )
13	11, 12	bitri 264	. . . 4 |- ( [. A / a ]. a = X <-> A = X )
14	3, 9, 13	3imtr3g 284	. . 3 |- ( [. A / a ]. ( Y R a -> a = X ) -> ( Y R A -> A = X ) )
15	2, 14	syl 17	. 2 |- ( A. a ( Y R a -> a = X ) -> ( Y R A -> A = X ) )
16		frege116.x	. . 3 |- X e. U
17		frege118.y	. . 3 |- Y e. V
18	16, 17	frege119 38276	. 2 |- ( ( A. a ( Y R a -> a = X ) -> ( Y R A -> A = X ) ) -> ( Fun `' `' R -> ( Y R X -> ( Y R A -> A = X ) ) ) )
19	15, 18	ax-mp 5	1 |- ( Fun `' `' R -> ( Y R X -> ( 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:  frege121  38278