Theorem dffrege115 36574

Description: If from the the circumstance that c is a result of an application of the procedure R to b, whatever b may be, it can be inferred that every result of an application of the procedure R to b is the same as c, then we say : "The procedure R is single-valued". Definition 115 of [Frege1879] p. 77. (Contributed by RP, 7-Jul-2020.)

Assertion

Ref	Expression
dffrege115	|- ( A. c A. b ( b R c -> A. a ( b R a -> a = c ) ) <-> Fun `' `' R )

Distinct variable group:   a, b, c, R

Proof of Theorem dffrege115

Step	Hyp	Ref	Expression
1		alcom 1923	. 2 |- ( A. c A. b ( b R c -> A. a ( b R a -> a = c ) ) <-> A. b A. c ( b R c -> A. a ( b R a -> a = c ) ) )
2		19.21v 1786	. . . . . . 7 |- ( A. a ( b R c -> ( b R a -> a = c ) ) <-> ( b R c -> A. a ( b R a -> a = c ) ) )
3		impexp 448	. . . . . . . . 9 |- ( ( ( b R c /\ b R a ) -> a = c ) <-> ( b R c -> ( b R a -> a = c ) ) )
4		vex 3048	. . . . . . . . . . . . 13 |- b e. _V
5		vex 3048	. . . . . . . . . . . . 13 |- c e. _V
6	4, 5	brcnv 5017	. . . . . . . . . . . 12 |- ( b `' `' R c <-> c `' R b )
7		df-br 4403	. . . . . . . . . . . 12 |- ( b `' `' R c <-> <. b , c >. e. `' `' R )
8	5, 4	brcnv 5017	. . . . . . . . . . . 12 |- ( c `' R b <-> b R c )
9	6, 7, 8	3bitr3ri 280	. . . . . . . . . . 11 |- ( b R c <-> <. b , c >. e. `' `' R )
10		vex 3048	. . . . . . . . . . . . 13 |- a e. _V
11	4, 10	brcnv 5017	. . . . . . . . . . . 12 |- ( b `' `' R a <-> a `' R b )
12		df-br 4403	. . . . . . . . . . . 12 |- ( b `' `' R a <-> <. b , a >. e. `' `' R )
13	10, 4	brcnv 5017	. . . . . . . . . . . 12 |- ( a `' R b <-> b R a )
14	11, 12, 13	3bitr3ri 280	. . . . . . . . . . 11 |- ( b R a <-> <. b , a >. e. `' `' R )
15	9, 14	anbi12ci 704	. . . . . . . . . 10 |- ( ( b R c /\ b R a ) <-> ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) )
16	15	imbi1i 327	. . . . . . . . 9 |- ( ( ( b R c /\ b R a ) -> a = c ) <-> ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
17	3, 16	bitr3i 255	. . . . . . . 8 |- ( ( b R c -> ( b R a -> a = c ) ) <-> ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
18	17	albii 1691	. . . . . . 7 |- ( A. a ( b R c -> ( b R a -> a = c ) ) <-> A. a ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
19	2, 18	bitr3i 255	. . . . . 6 |- ( ( b R c -> A. a ( b R a -> a = c ) ) <-> A. a ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
20	19	albii 1691	. . . . 5 |- ( A. c ( b R c -> A. a ( b R a -> a = c ) ) <-> A. c A. a ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
21		alcom 1923	. . . . 5 |- ( A. c A. a ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) <-> A. a A. c ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
22	20, 21	bitri 253	. . . 4 |- ( A. c ( b R c -> A. a ( b R a -> a = c ) ) <-> A. a A. c ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
23		opeq2 4167	. . . . . 6 |- ( a = c -> <. b , a >. = <. b , c >. )
24	23	eleq1d 2513	. . . . 5 |- ( a = c -> ( <. b , a >. e. `' `' R <-> <. b , c >. e. `' `' R ) )
25	24	mo4 2346	. . . 4 |- ( E* a <. b , a >. e. `' `' R <-> A. a A. c ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
26		mo2v 2306	. . . 4 |- ( E* a <. b , a >. e. `' `' R <-> E. c A. a ( <. b , a >. e. `' `' R -> a = c ) )
27	22, 25, 26	3bitr2i 277	. . 3 |- ( A. c ( b R c -> A. a ( b R a -> a = c ) ) <-> E. c A. a ( <. b , a >. e. `' `' R -> a = c ) )
28	27	albii 1691	. 2 |- ( A. b A. c ( b R c -> A. a ( b R a -> a = c ) ) <-> A. b E. c A. a ( <. b , a >. e. `' `' R -> a = c ) )
29		relcnv 5207	. . . 4 |- Rel `' `' R
30	29	biantrur 509	. . 3 |- ( A. b E. c A. a ( <. b , a >. e. `' `' R -> a = c ) <-> ( Rel `' `' R /\ A. b E. c A. a ( <. b , a >. e. `' `' R -> a = c ) ) )
31		dffun5 5595	. . 3 |- ( Fun `' `' R <-> ( Rel `' `' R /\ A. b E. c A. a ( <. b , a >. e. `' `' R -> a = c ) ) )
32	30, 31	bitr4i 256	. 2 |- ( A. b E. c A. a ( <. b , a >. e. `' `' R -> a = c ) <-> Fun `' `' R )
33	1, 28, 32	3bitri 275	1 |- ( A. c A. b ( b R c -> A. a ( b R a -> a = c ) ) <-> Fun `' `' R )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1442   E.wex 1663   e. wcel 1887   E*wmo 2300   <.cop 3974   class class class wbr 4402   `'ccnv 4833   Rel wrel 4839   Fun wfun 5576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-fun 5584

This theorem is referenced by:  frege116  36575